# Around Furstenberg’s times $p$, times $q$ conjecture: times $p$-invariant
measures with some large Fourier coefficients

Catalin Badea  Univ. Lille, CNRS, UMR 8524 - Laboratoire Paul Painlevé,
F-59000 Lille, France<EMAIL_ADDRESS>Sophie Grivaux  CNRS, Univ.
Lille, UMR 8524 - Laboratoire Paul Painlevé, F-59000 Lille, France
<EMAIL_ADDRESS>
###### Abstract

For each integer $n\geq 1$, denote by $T_{n}$ the map $x\mapsto nx\mod 1$ from
the circle group $\mathbb{T}=\mathbb{R}/\mathbb{Z}$ into itself. Let $p,q\geq
2$ be two multiplicatively independent integers. Using Baire Category
arguments, we show that generically a $T_{p}$-invariant probability measure
$\mu$ on $\mathbb{T}$ with no atom has some large Fourier coefficients along
the sequence $(q^{n})_{n\geq 0}$. In particular, $(T_{q^{n}}\mu)_{n\geq 0}$
does not converges weak-star to the normalised Lebesgue measure on
$\mathbb{T}$. This disproves a conjecture of Furstenberg and complements
previous results of Johnson and Rudolph. In the spirit of previous work by
Meiri and Lindenstrauss-Meiri-Peres, we study generalisations of our main
result to certain classes of sequences $(c_{n})_{n\geq 0}$ other than the
sequences $(q^{n})_{n\geq 0}$, and also investigate the multidimensional
setting.

###### keywords:

$\times_{p}$-invariant measures; Furstenberg Conjecture; Fourier coefficients
of continuous measures; Baire Category methods

###### 1991 Mathematics Subject Classification:

43A25, 37A05, 54E52, 37A25

C.B. and S.G. were supported in part by the project FRONT of the French
National Research Agency (grant ANR-17-CE40-0021) and by the Labex CEMPI
(grant ANR-11-LABX-0007-01). C.B. was supported in part by Max Planck
Institute for Mathematics in Bonn.

##  1\. Introduction and main results

###  1.a. Synopsis

In the late 1960s, Furstenberg proved significant results and proposed
fascinating conjectures that aimed to express in various ways the heuristic
principle that expansions in multiplicatively independent bases have no shared
structure. For further details about this idea, readers can refer to the
recent survey [Shm] which also outlines some progress in Furstenberg’s
programme. Here, we shall list one result and three conjectures due to
Furstenberg; some known partial results related to these conjectures will be
mentioned in the following subsection. _In all these statements,_ $p,q\geq 2$
_are two fixed multiplicatively independent integers_. Recall that $p,q\geq 2$
are called _multiplicatively independent_ if $\log p/\log q\not\in\mathbb{Q}$.
For each integer $n\geq 1$, denote by $T_{n}$ the map $x\mapsto nx\mod 1$ from
the circle group $\mathbb{T}=\mathbb{R}/\mathbb{Z}$, identified with $[0,1)$,
into itself. A subset $F$ of $\mathbb{T}$ is said to be $T_{n}$-invariant if
$T_{n}(F)\subset F$. Notice that $T_{n}$ shifts the $n$-ary expansion of a
real number and that each map $T_{n}$ has many closed, infinite invariant
subsets.

The following topological rigidity result has been proved in [F].

######  Theorem 1.1 (Furstenberg).

The only infinite closed subset $F$ of $\mathbb{T}$ which is simultaneously
$T_{p}$\- and $T_{q}$-invariant is $F=\mathbb{T}$.

Furstenberg formulated a conjecture, called Conjecture (C1) in [L2], which is
stronger than Theorem 1.1 and deals with the asymptotic behaviour of a
$T_{p}$–invariant subset under the action of $T_{q}$.

######  Conjecture 1.2.

Let $F$ be an infinite closed $T_{p}$-invariant subset. Then the iterates
$T_{q}^{n}(F)$ converge in the Hausdorff distance to $\mathbb{T}$ as $n$ tends
to infinity.

The measure-theoretical analogue statements of Theorem 1.1 and Conjecture 1.2
are both conjectural statements. Recall that a Borel probability measure $\mu$
on $\mathbb{T}$ is said to be $T_{n}$-invariant if $\mu=T_{n}\mu$, where
$T_{n}\mu$ is the measure defined by $T_{n}\mu(C)=\mu(T_{n}^{-1}C)$ for every
measurable set $C$. There are uncountable many $T_{n}$-invariant, or even
ergodic, measures; see for instance [BenQui, p. 141]. The next measure-
theoretical rigidity conjecture, called Conjecture (C2) in [L2], is the
renowned $\times p$, $\times q$ conjecture of Furstenberg, one of the most
fundamental open questions in ergodic theory. We say that a probability
measure is _continuous_ if it has no atom.

######  Conjecture 1.3 ($\times p$, $\times q$ conjecture).

The only continuous Borel probability measure on $\mathbb{T}$ which is
simultaneously $T_{p}$\- and $T_{q}$-invariant is the (normalised) Lebesgue
measure _Leb_.

The natural analogue of Conjecture 1.2 for measures was also conjectured by
Furstenberg: this is Conjecture (C3) in [L2] and concerns now the convergence
in the weak-star topology of a $T_{p}$-invariant measure under the action of
$T_{q}$.

######  Conjecture 1.4.

Let $\mu$ be a continuous Borel probability measure on $\mathbb{T}$ which is
$T_{p}$-invariant. Then $T_{q}^{n}\mu$ converge $w^{*}$ to _Leb_.

It is easy to see that Conjecture 1.4 implies the $\times p$, $\times q$
conjecture (Conjecture 1.3): suppose indeed that Conjecture 1.4 is true, and
let $\mu$ be a continuous and simultaneously $T_{p}$\- and $T_{q}$-invariant
measure as in Conjecture 1.3. Since
$T_{q}^{n}(\mu)={T_{q^{n}}}\mu\xrightarrow{\text{$w^{*}$}}\textrm{Leb}$, the
Fourier coefficients of $\mu$ verify $\widehat{\mu}(aq^{n})\rightarrow 0$ for
every $a\in\mathbb{Z}\setminus\\{0\\}$. Since $\mu$ is $T_{q}$-invariant, this
implies that $\widehat{\mu}(a)=0$ for every $a\in\mathbb{Z}\setminus\\{0\\}$,
so that $\mu=\textrm{Leb}$.

The main aim of this manuscript is to show that generically (in the Baire
Category sense), a continuous $T_{p}$-invariant probability measure $\mu$ on
$\mathbb{T}$ has some large Fourier coefficients along the sequence
$(q^{n})_{n\geq 0}$. This implies that the sequence $(T_{q}^{n}(\mu))_{n\geq
0}=(T_{q^{n}}\mu)_{n\geq 0}$ does not converge $w^{*}$ to Leb, _disproving_
thus Conjecture 1.4. The precise statement is given in Theorem 1.5 below.

It follows from our results and some results of Johnson and Rudolph in [JR]
that generically, in the Baire Category sense, $(T_{q^{n}}\mu)_{n\geq 0}$ does
not converge $w^{*}$ to the Lebesgue measure, but the convergence of
$T_{q^{n}}\mu$ to Leb holds along a “large” sequence of integers (a sequence
of upper density $1$); see Corollary 1.6. This sheds some light on the
complexity of the asymptotic behaviour of the action of $T_{q}$ on a generic
$T_{p}$-invariant measure. In the spirit of previous work by Meiri [M] and
Lindenstrauss-Meiri-Peres [LMP], we study generalisations of our main result
to certain classes of sequences $(c_{n})_{n\geq 0}$ other than the sequences
$(q^{n})_{n\geq 0}$, and also investigate the multidimensional setting.

Our methods are mainly functional-analytic, based for instance on Baire
category methods and the Hahn-Banach theorem. We also use tools from classical
harmonic analysis (Fourier coefficients of measures, $p$-Bernoulli measures),
ergodic theory (the periodic specification property, the ergodic decomposition
theorem) and elementary number theory.

###  1.b. Background

Without claiming completeness, we mention some previous contributions related
to Theorem 1.1 and Conjectures 1.2, 1.3 and 1.4.

Many aspects of the dynamics of subsemigroups of $(T_{n})_{n\geq 1}$ were
discussed in the seminal paper [F] by Furstenberg. The proof of Theorem 1.1 in
[F] used the disjointness of specific dynamical systems, a notion introduced
in [F]. An elementary proof of Theorem 1.1 has been given by Boshernitzan
[Bosh] and an “effective” version has been proved in [BLMV] by Bourgain,
Lindenstrauss, Michel and Venkatesh. Starting with Berend [Ber], several
authors studied multidimensional generalisations of Theorem 1.1.

Conjecture 1.2 is largely open. It is known that if $F$ is a $T_{p}$-invariant
subset of $\mathbb{T}$, then there exists a subsequence $(q^{n_{k}})$ such
that $T_{q}^{n_{k}}(F)$ converges to $\mathbb{T}$ in the Hausdorff metric; see
for instance [Kra, Lemma 2.1]. Another result related to Conjecture 1.2 can be
found in [MP, Th. 1.1]. Starting with the papers [BerPerJLMS, AlonPeres] by
Berend-Peres and Alon-Peres, several authors studied the so-called Glasner
sets. A set $S$ of integers is said to be a _Glasner set_ if for every
infinite closed subset $F$ of $\mathbb{T}$, there exists a sequence $(c_{n})$
of elements in $S$ such that $T_{c_{n}}(F)$ converges to $\mathbb{T}$ in the
Hausdorff metric. With this terminology, a result from [Glasner] can be
formulated as the fact that the set of integers is a Glasner set. Other quite
small sets of integers are Glasner, like sets of positive (Banach) density or
the sets of values assumed by any non-constant polynomial mapping the natural
numbers to themselves. Note however that a finite union of lacunary sequences
is not a Glasner set ([BerPerJLMS, Th. 1.4]). Glasner sets have been also
studied in the multidimensional setting.

The first result about the $\times p$, $\times q$ conjecture has been proved
by Lyons in [L2], the first place where Conjecture 1.3 appeared in print: if
$p$ and $q$ are relatively prime, any probability measure on $\mathbb{T}$
which is $T_{p}$\- and $T_{q}$-invariant and $T_{p}$-exact (i.e. has
completely positive entropy with respect to $T_{p}$), must be the Lebesgue
measure. Rudolph substantially strengthened this theorem in [R], showing that
the conclusion is true with only the weaker assumption that the measure is
ergodic under the joint action of $T_{p}$ and $T_{q}$, and of positive entropy
under the action of $T_{p}$. Johnson [J] then generalised this to the case
where $p$ and $q$ are multiplicatively independent. A different argument,
along the lines of Lyons [L2], was given by Feldman [Feldman]. Other different
proofs were given by Host [H] and Parry [Parry]. In all these proofs the
positive entropy remains a crucial assumption. The Rudolph-Johnson theorem has
been used by Einsiedler and Fish [EinsFish] to prove that a continuous Borel
probability measure on $\mathbb{T}$ invariant under the action of a
multiplicative semigroup with positive lower logarithmic density is the
normalised Lebesgue measure. An important advance was made by Katok and
Spatzier [KS96], who discovered that Rudolph’s proof can be extended to give
partial information on invariant measures in much greater generality. We also
mention the works [Elon-padic, HochShm, Hoch, Hochman], as well as the surveys
[Lind, EnsLind, LindMarg], for an account of recent progress on measure
rigidity for higher rank diagonal actions on homogeneous spaces.

Some partial results about Conjecture 1.4 (conjecture (C3) in [L2]), which
will be disproved in this manuscript, are also known. The study of convergence
of the sequence $(T_{q^{n}}\mu)_{n\geq 0}$ to the Lebesgue measure for certain
classes of $T_{p}$-invariant measures $\mu$ lies at the core of the works of
Lyons [L1, L2], Feldman and Smorodinsky [FS], Johnson and Rudolph [JR], and
Host [H]. Given $p,q\geq 2$ two multiplicatively independent integers, it is
shown in [L1] (see also [L2]) that if $\mu$ is a non-degenerate $p$-Bernoulli
measure, then $T_{q^{n}}\mu\xrightarrow{\text{$w^{*}$}}\textrm{Leb}$. The main
result of [FS] states that under the same assumption, $\mu$-almost every
$x\in[0,1]$ is normal to the base $q$. It is proved by Host in [H] that
whenever $p$ and $q$ are relatively prime, any measure
$\mu\in\mathcal{P}_{p}(\mathbb{T})$ which is ergodic and has positive entropy
with respect to $T_{p}$ is such that $\mu$-almost every $x\in[0,1]$ is normal
to the base $q$. The generalisation to the case where $p$ and $q$ are
multiplicatively independent was obtained by Hochman and Shmerkin [HochShm].
The Host-Hochman-Shmerkin result implies easily a result by Johnson and
Rudolph [JR] that for every such ergodic with positive entropy measure
$\mu\in\mathcal{P}_{p}(\mathbb{T})$,

$\dfrac{1}{N}\sum_{n=0}^{N-1}T_{q^{n}}\mu\xrightarrow{\text{$w^{*}$}}\textrm{Leb}.$

Johnson and Rudolph observe the following consequence: if
$\mu\in\mathcal{P}_{p}(\mathbb{T})$ is ergodic and of positive entropy with
respect to $T_{p}$, then
$T_{q^{n}}\mu\xrightarrow{\text{$w^{*}$}}\textrm{Leb}$ on a sequence of Banach
density one (called a sequence of _uniform full density_ in [JR]). As a
consequence, they obtain that the set

$G^{\prime}_{p,q}:=\\{\mu\in\mathcal{P}_{p,c}(\mathbb{T})\;;\;{T_{q^{n}}\mu\xrightarrow{\text{$w^{*}$}}\textrm{Leb}}\
\textrm{along a sequence of upper density 1}\\}$

is residual in $(\mathcal{P}_{p}(\mathbb{T}),w^{*})$. So, generically in the
Baire Category sense, convergence of $T_{q^{n}}\mu$ to the Lebesgue measure
holds along a “large” sequence of integers. But the Baire Category arguments
leave room for possible “bad” sequences where convergence to the Lebesgue
measure, as predicted by Conjecture 1.4, cannot be guaranteed. Quoting from
[JR]: “ _As we have no examples showing such bad sequences can actually exist,
perhaps it is possible by some more explicit investigation to eliminate these
bad sequences along which convergence to the Lebesgue measure fails_ ”. Our
first main result, which is Theorem 1.5 below, shows that generically such bad
sequences do exist, and cannot be eliminated.

###  1.c. Notation

Denote by $\mathcal{P}(\mathbb{T})$ the space of Borel probability measures on
$\mathbb{T}$, and, for any $p\geq 2$, by $\mathcal{P}_{p}(\mathbb{T})$ the
space of $T_{p}$-invariant measures $\mu\in\mathcal{P}(\mathbb{T})$. We endow
$\mathcal{P}(\mathbb{T})$ with the topology of $w^{*}$-convergence of
measures, which turns it into a compact metrizable space. Recall that given
measures $\mu_{k}$, $k\geq 1$, and $\mu$ belonging to
$\mathcal{P}(\mathbb{T})$, we say that
$\mu_{k}\xrightarrow{\text{$w^{*}$}}\mu$ if

$\displaystyle\int_{\mathbb{T}}fd\mu_{k}\longrightarrow\int_{\mathbb{T}}fd\mu$

as $k\longrightarrow+\infty$ for every $f\in C(\mathbb{T})$, where
$C(\mathbb{T})$ is the space of continuous functions on $\mathbb{T}$, endowed
with the sup norm $||\,.\,||_{\infty,\mathbb{T}}$ on $\mathbb{T}$. This is
equivalent to requiring that $\hat{\mu}_{k}(a)\longrightarrow\hat{\mu}(a)$ for
every $a\in\mathbb{Z}$, where the $a$-th Fourier coefficient of a measure
$\nu\in\mathcal{P}(\mathbb{T})$ is defined in this manuscript as

$\hat{\nu}(a)=\int_{\mathbb{T}}z^{a}d\nu(z).$

We denote by $\mathcal{P}_{c}(\mathbb{T})$ the set of continuous (i.e. non-
atomic) measures on $\mathbb{T}$, and by $\mathcal{P}_{p,c}(\mathbb{T})$ the
set of continuous $T_{p}$-invariant measures on $\mathbb{T}$. Since
$\mathcal{P}_{p}(\mathbb{T})$ is $w^{*}$-closed in $\mathcal{P}(\mathbb{T})$,
$(\mathcal{P}_{p}(\mathbb{T}),w^{*})$ is also a compact metrizable space. In
particular, $(\mathcal{P}_{p}(\mathbb{T}),w^{*})$ is a Polish space, in which
the Baire Category Theorem applies. Recall that a subset of a Polish space is
called _residual_ if it contains a dense $G_{\delta}$ set (i.e. a countable
intersection of dense open sets).

For our study of the multidimensional setting the following notation is
required. For each $d\geq 2$, we denote by $\mathcal{P}(\mathbb{T}^{d})$ the
set of Borel probability measures on $\mathbb{T}^{d}$, and by
$\mathcal{P}_{c}(\mathbb{T}^{d})$ the set of continuous measures
$\mu\in\mathcal{P}(\mathbb{T}^{d})$. Given a matrix $A\in M_{d}(\mathbb{Z})$
with $\det(A)\neq 0$, we denote by $T_{A}$ the associated transformation
$\boldsymbol{x}\mapsto A\boldsymbol{x}\mod 1$ of $\mathbb{T}^{d}$ into itself.
This transformation preserves the normalised Lebesgue measure on
$\mathbb{T}^{d}$, which we write as $\textrm{Leb}_{d}$. Notice that $T_{A}$ is
an ergodic transformation of $(\mathbb{T}^{d},\textrm{Leb}_{d})$ if and only
if no eigenvalue of $A$ is a root of unity. The set of $T_{A}$-invariant
measures on $\mathbb{T}^{d}$ is denoted by $\mathcal{P}_{A}(\mathbb{T})$, and
$\mathcal{P}_{A,c}(\mathbb{T})$ is the set of continuous $T_{A}$-invariant
measures on $\mathbb{T}^{d}$.

###  1.d. Main results

Here is our first main result, showing that generically a continuous
$T_{p}$-invariant probability measure $\mu$ on $\mathbb{T}$ has some large
Fourier coefficients along the sequence $(q^{n})_{n\geq 0}$.

######  Theorem 1.5 (large Fourier coefficients).

Let $p,q\geq 2$ be two distinct integers. Then the set

$S_{p,q}:=\bigl{\\{}\mu\in\mathcal{P}_{p,c}(\mathbb{T})\;;\;\limsup\limits_{n\to+\infty}|\widehat{\mu}(q^{n})|>0\bigr{\\}}$

is residual in $(\mathcal{P}_{p}(\mathbb{T}),w^{*})$. In particular, the set

$G_{p,q}:=\\{\mu\in\mathcal{P}_{p,c}(\mathbb{T})\;;\;T_{q^{n}}\mu\lx@xy@svg{\hbox{\raise
0.0pt\hbox{\kern
3.0pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
6.40012pt\raise 4.97223pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.97223pt\hbox{$\scriptstyle{w^{*}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 23.0pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces{\hbox{\kern
11.05365pt\raise-0.1pt\hbox{\hbox{\kern
0.0pt\raise-1.75pt\hbox{$\textstyle{\scriptstyle/}$}}}}}{\hbox{\kern
23.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}\ignorespaces}}}}\ignorespaces\emph{Leb}\
\textrm{as}\ n\to+\infty\\}$

is residual in $(\mathcal{P}_{p}(\mathbb{T}),w^{*})$, thus disproving
Conjecture 1.4.

We should note that the proof of Theorem 1.5 does not require that $p$ and $q$
be multiplicatively independent; if $p$ and $q$ are powers of the same
integer, then a simple and direct proof of Theorem 1.5 can be given.

By combining Theorem 1.5 with the results presented in [JR], we can derive the
following corollary.

######  Corollary 1.6.

Let $p,q\geq 2$ be two multiplicatively independent integers. Then the set of
all measures $\mu\in\mathcal{P}_{p,c}(\mathbb{T})$ such that

$T_{q^{n}}\mu\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
3.0pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
6.40012pt\raise 4.97223pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.97223pt\hbox{$\scriptstyle{w^{*}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 23.0pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces{\hbox{\kern
11.05365pt\raise-0.1pt\hbox{\hbox{\kern
0.0pt\raise-1.75pt\hbox{$\textstyle{\scriptstyle/}$}}}}}{\hbox{\kern
23.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}\ignorespaces}}}}\ignorespaces\emph{Leb}\
\textrm{as}\ n\to+\infty$

and

${T_{q^{n}}\mu\xrightarrow{\text{$w^{*}$}}\emph{Leb}}\ \textrm{along a
sequence of upper density 1}$

is residual in $(\mathcal{P}_{p}(\mathbb{T}),w^{*})$.

Meiri [M] and Lindenstrauss, Meiri and Peres [LMP] generalised the results
from [H] and [JR] to certain classes of sequences $(c_{n})_{n\geq 0}$ other
than the sequences $(q^{n})_{n\geq 0}$. More precisely ([M]), if the sequence
of remainders $(c_{n}\mod p^{N})_{0\leq n<p^{N}}$, $N\geq 1$, satisfies
certain combinatorial properties, then every $T_{p}$-invariant ergodic measure
$\mu$ of positive entropy is such that $(c_{n}x)_{n\geq 0}$ is uniformly
distributed mod $1$ for $\mu$-almost every $x\in[0,1]$. A weaker combinatorial
condition on the sequence $(c_{n})_{n\geq 0}$ is introduced in [LMP]: if the
so-called $p$_-adic collision exponent_ $\Gamma_{p}((c_{n}))$ is less that
$2$, then every measure $\mu\in\mathcal{P}_{p}(\mathbb{T})$ which is ergodic
and of positive entropy is $(c_{n})$_-generic_ in the sense that

$\dfrac{1}{N}\sum_{n=0}^{N-1}T_{c_{n}}\mu\xrightarrow{\text{$w^{*}$}}\textrm{Leb}.$

It follows that the set of measures $\mu\in\mathcal{P}_{p,c}(\mathbb{T})$ such
that $T_{c_{n}}\mu\xrightarrow{\text{$w^{*}$}}\textrm{Leb}$ along a sequence
of upper density $1$ is residual in $(\mathcal{P}_{p}(\mathbb{T}),w^{*})$.

Conjecture 1.4 thus fits in a much broader framework: given a strictly
increasing sequence of integers $(c_{n})_{n\geq 0}$ of integers, is it true
that the set

$G_{p,(c_{n})}:=\bigl{\\{}\mu\in\mathcal{P}_{p,c}(\mathbb{T})\;;\;T_{c_{n}}\mu\lx@xy@svg{\hbox{\raise
0.0pt\hbox{\kern
3.0pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
6.40012pt\raise 4.97223pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.97223pt\hbox{$\scriptstyle{w^{*}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 23.0pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces{\hbox{\kern
11.05365pt\raise-0.1pt\hbox{\hbox{\kern
0.0pt\raise-1.75pt\hbox{$\textstyle{\scriptstyle/}$}}}}}{\hbox{\kern
23.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}\ignorespaces}}}}\ignorespaces\textrm{Leb}\quad\textrm{as}\
{n}\longrightarrow{+\infty}\bigr{\\}}$

is residual in $(\mathcal{P}_{p}(\mathbb{T}),w^{*})$? We prove in Theorem 2.1
below a very general criterion on the sequence $(c_{n})_{n\geq 0}$ implying an
affirmative answer to this question. It allows to deal with most of the
classes of sequences considered in [M] and [LMP], and we obtain for instance
the following theorem, which complements [LMP, Th. 1.4] and [M, Th. B]:

######  Theorem 1.7 (linear recurrent sequences as $(c_{n})$).

Let $(c_{n})_{n\geq 0}$ be a sequence of integers satisfying a linear
recursion of the form

$c_{n}=a_{1}c_{n-1}+a_{2}c_{n-2}+\cdots+a_{L}c_{n-L},\quad n>L$

for some $L\geq 1$ and integer coefficients $a_{1},\dots,a_{L}$ with
$a_{L}\neq 0$. If the integers $a_{L}$ and $p$ are relatively prime, then the
set

$G_{p,(c_{n})}=\bigl{\\{}\mu\in\mathcal{P}_{p,c}(\mathbb{T})\;;\;T_{c_{n}}\mu\lx@xy@svg{\hbox{\raise
0.0pt\hbox{\kern
3.0pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
6.40012pt\raise 4.97223pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.97223pt\hbox{$\scriptstyle{w^{*}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 23.0pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces{\hbox{\kern
11.05365pt\raise-0.1pt\hbox{\hbox{\kern
0.0pt\raise-1.75pt\hbox{$\textstyle{\scriptstyle/}$}}}}}{\hbox{\kern
23.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}\ignorespaces}}}}\ignorespaces\emph{Leb}\quad\emph{as}\
{n}\longrightarrow{+\infty}\bigr{\\}}$

is residual in $(\mathcal{P}_{p}(\mathbb{T}),w^{*})$.

Notice that when $L=1$ and $a_{1}=q$, Theorem 1.7 reduces to the case studied
in Theorem 1.5, but with the additional requirement that $p$ and $q$ are
relatively prime.

We also mention the following related result. In [BadGri], a continuous
probability measure $\mu$ on $\mathbb{T}$ was constructed with the property
that for any increasing sequence $(c_{n})$ in the multiplicative semigroup
$\\{p^{m}q^{n}:m,n\geq 0\\}$, one has $T_{c_{n}}\mu\lx@xy@svg{\hbox{\raise
0.0pt\hbox{\kern
3.0pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
6.40012pt\raise 4.97223pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.97223pt\hbox{$\scriptstyle{w^{*}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 23.0pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces{\hbox{\kern
11.05365pt\raise-0.1pt\hbox{\hbox{\kern
0.0pt\raise-1.75pt\hbox{$\textstyle{\scriptstyle/}$}}}}}{\hbox{\kern
23.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}\ignorespaces}}}}\ignorespaces\textrm{Leb}$.
This disproved Conjectures (C4) and (C5) from [L2]. However, it appears that
the construction from [BadGri] cannot be modified to produce a measure that is
$T_{p}$-invariant.

We now move over to the multidimensional setting. The equidistribution result
[H] of Host was generalised to the multidimensional setting by Meiri-Peres
[MP], Host [H2] himself and Algom [Alg]. The general framework of these works
is the following: given two endomorphisms $A$ and $B$ of $\mathbb{T}^{d}$ and
a measure $\mu\in\mathcal{P}_{A}(\mathbb{T}^{d})$, study the equidistribution
properties of the sequence $(B^{n}\boldsymbol{x})_{n\geq 0}$ for $\mu$-almost
every $\boldsymbol{x}\in\mathbb{T}^{d}$. This problem is studied in [H2] when
$\mu$ is $A$-ergodic and has positive entropy, under the condition that
$\det(A)$ and $\det(B)$ are relatively prime (which is exactly condition (b)
of Theorem 1.8 below), plus some other assumptions on matrices $A$ and $B$. It
is proved in [H2] that for every ergodic measure
$\mu\in\mathcal{P}_{A}(\mathbb{T}^{d})$ of positive entropy, the sequence
$(B^{n}\boldsymbol{x})_{n\geq 0}$ is uniformly distributed in $\mathbb{T}^{d}$
for $\mu$-almost every $\boldsymbol{x}\in\mathbb{T}^{d}$. The paper [MP]
considers the case where $A$ and $B$ are both diagonal matrices,
$A=\textrm{diag}(a_{1},\dots,a_{d})$, $B=\textrm{diag}(b_{1},\dots,b_{d})$,
with $|a_{i}|>1$, $|b_{i}|>1$, and $\textrm{gcd}(a_{i},b_{i})=1$ for every
$i\in\\{1\ldots d\\}$.

Accordingly, we complement these results by showing a multidimensional version
of Theorem 1.5.

######  Theorem 1.8 (multidimensional setting).

Let $d\geq 2$ and let $A,B\in M_{d}(\mathbb{Z})$ with $\det(A)\neq 0$ and
$\det(B)\neq 0$. Suppose that

  1. _(a)_

A is similar to a diagonal matrix
$D=\emph{diag}(\lambda_{1},\dots,\lambda_{d})$, where $|\lambda_{j}|\neq 1$,
$1\leq j\leq d$;

and either

  1. _(b)_

$\det(A)$ and $\det(B)$ are relatively prime;

or

  1. _(b’)_

$A$ is upper or lower triangular.

Then the set

$G_{A,B}{}:=\bigl{\\{}\mu\in\mathcal{P}_{A,c}(\mathbb{T}^{d})\;;\;T_{B^{n}}\mu\lx@xy@svg{\hbox{\raise
0.0pt\hbox{\kern
3.0pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 3.0pt\raise
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6.40012pt\raise 4.97223pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.97223pt\hbox{$\scriptstyle{w^{*}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 23.0pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces{\hbox{\kern
11.05365pt\raise-0.1pt\hbox{\hbox{\kern
0.0pt\raise-1.75pt\hbox{$\textstyle{\scriptstyle/}$}}}}}{\hbox{\kern
23.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}\ignorespaces}}}}\ignorespaces\emph{Leb}_{d}\bigr{\\}}$

is residual in $(\mathcal{P}_{A}(\mathbb{T}^{d}),w^{*})$.

###  1.e. Overview

The paper is organised as follows. We present in Section 2 a general criterion
on a sequence $(c_{n})_{n\geq 0}$ of integers implying that the set
$G_{p,(c_{n})}$ is residual in $(\mathcal{P}_{p}(\mathbb{T}),w^{*})$. This
criterion is the object of Theorem 2.1. Its proof relies on a density result
for certain classes of discrete measures in
$(\mathcal{P}_{p}(\mathbb{T}),w^{*})$ (Theorem 2.3), which is of interest in
itself and involves the so-called periodic specification property of the
transformation $T_{p}$. We present in Section 3 various examples of sequences
considered in [M] and [LMP] which satisfy the assumptions of Theorem 2.1, and
derive Theorems 1.5 and 1.7 from Theorem 2.1. The multidimensional case is
treated in Section 4. Since assumption (a) of Theorem 1.8 does not necessarily
imply that $T_{A}:\mathbb{T}^{d}\longrightarrow\mathbb{T}^{d}$ has the
periodic specification property, we need a different argument (Theorem 4.1) in
order to show the density in $(\mathcal{P}_{A}(\mathbb{T}^{d}),w^{*})$ of the
relevant classes of $T_{A}$-invariant measures. We discuss in Section 5 a
different approach to the Johnson-Rudolph result of [JR] that the set

$G\,^{\prime}_{\\!p,q}:=\\{\mu\in\mathcal{P}_{p,c}(\mathbb{T})\;;\;{T_{q^{n}}\mu\xrightarrow{\text{$w^{*}$}}\textrm{Leb}}\
\textrm{along a sequence of upper density 1}\\}$

is residual in $(\mathcal{P}_{p}(\mathbb{T}),w^{*})$ for multiplicatively
independent integers $p$ and $q$, and present some related results and open
questions.

##  2\. Classes of $\times_{p}$-invariant measures with some large Fourier
coefficients

In the whole section, $p\geq 2$ will be a fixed integer. Let $(c_{n})_{n\geq
0}$ be a strictly increasing sequence of integers. We say that $(c_{n})_{n\geq
0}$ satisfies _assumption_ (H) if the following is true:

(H)  There exist finitely many nonnegative integers
$t_{1},\dots,t_{r},h_{1},\dots,h_{d}$ with $h_{l}\neq 0$ for every
$l\in\\{1,\ldots,d\\}$, and an infinite subset $I$ of $\mathbb{N}$ such that
for every $N\in I$, there exist $i\in\\{1,\dots,r\\}$ and
$l\in\\{1,\dots,d\\}$ with the property that $h_{l}c_{n}\equiv
t_{i}\mod(p^{N}-1)$ for infinitely many integers $n$.

Our aim in this section is to prove the following theorem:

######  Theorem 2.1.

Let $(c_{n})_{n\geq 0}$ be a strictly increasing sequence of integers
satisfying assumption _(H)_. Then the set

$G_{p,(c_{n})}=\bigl{\\{}\mu\in\mathcal{P}_{p,c}(\mathbb{T})\;;\;T_{c_{n}}\mu\lx@xy@svg{\hbox{\raise
0.0pt\hbox{\kern
3.0pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
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0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
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3.0pt}}}}}}\ignorespaces{\hbox{\kern 23.0pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces{\hbox{\kern
11.05365pt\raise-0.1pt\hbox{\hbox{\kern
0.0pt\raise-1.75pt\hbox{$\textstyle{\scriptstyle/}$}}}}}{\hbox{\kern
23.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}\ignorespaces}}}}\ignorespaces\emph{{Leb}}\quad\textrm{as
}n\longrightarrow+\infty\bigr{\\}}$

is residual in $(\mathcal{P}_{p}(\mathbb{T}),w^{*})$.

Condition (H) may look somewhat technical, but it is actually a rather weak
one. We shall exhibit in Section 3 many examples of sequences $(c_{n})_{n\geq
0}$ satisfying (H). In particular, the sequence $(c_{n})=(q^{n})$ satisfies it
for any $q\geq 2$. This disproves Conjecture 1.4.

Assumption (H) is of the same nature as the congruence assumptions mod $p^{N}$
which appear in the works of Host [H] and Meiri [M], and which are formalised
in terms of $p$-adic collision exponent in [LMP]. These two assumptions are
nonetheless different, be it only because (H) involves congruences mod
$(p^{N}-1)$, while the $p$-adic collision exponent is defined in terms of
congruence mod $p^{N}$.

Our main tool for the proof of Theorem 2.1 is a density result for certain
families of discrete $T_{p}$-invariant measures on $\mathbb{T}$.

###  2.a. Density of discrete $\times_{p}$-invariant measures

The periodic points of the transformation $T_{p}$ are exactly the points
$\lambda\in\mathbb{T}$ such that $\lambda^{p^{N}}=\lambda$ for some $N\geq 1$.
In this case, the probability measure $\mu_{\lambda}$ on $\mathbb{T}$, defined
as

$\mu_{\lambda}=\dfrac{1}{N}\sum_{j=0}^{N-1}\delta_{\\{\lambda^{p^{j}}\\}},$

is a discrete $T_{p}$-invariant measure on $\mathbb{T}$ whose support is the
orbit of the point $\lambda$ under the action of $T_{p}$. It is ergodic for
$T_{p}$, and the set of all such measures (where $\lambda$ varies over the set
of all $(p^{N}-1)$-th roots of 1, $N\geq 1$) is dense in
$(\mathcal{P}_{p}(\mathbb{T}),w^{*})$ [S1, S2].

This density property is deeply linked to the fact that the dynamical system
$(\mathbb{T},T_{p})$ has the so-called _specification property_ introduced by
Bowen in [B] (see also [S1, S2]). Since it will be needed in the sequel, we
recall here the definition from [S1]. The setting is that of compact dynamical
systems $(X,T)$, where $(X,d)$ is a compact metric space and $T$ is a
continuous self-map of $X$. This property is often referred to as the
_periodic specification property_ , and it is the terminology we shall use
here. The article [KLO] contains an overview of the specification property and
its many variants.

######  Definition 2.2.

The system $(X,T)$ is said to have the _periodic specification property_ if
for every $\varepsilon>0$ there exists $N_{\varepsilon}\in\mathbb{N}$ such
that for every integers $0\leq a_{1}\leq b_{1}$ and $0\leq a_{2}\leq b_{2}$
with $a_{2}-b_{1}>N_{\varepsilon}$, for every vectors $x_{1},x_{2}\in X$, and
for every integer $d>b_{2}-a_{1}+N_{\varepsilon}$, there exists a periodic
point $x$ for $T$ with period $d$ such that

  1. (i)

$d(T^{j}x,T^{j}x_{1})<\varepsilon$ for every $j=a_{1},\ldots,b_{1}$;

  2. (ii)

$d(T^{j}x,T^{j}x_{2})<\varepsilon$ for every $j=a_{2},\ldots,b_{2}$.

If $x$ is periodic for $T$ with period $d$, the measure

$\mu_{x}=\dfrac{1}{d}\displaystyle\sum_{j=0}^{d-1}\delta_{\\{T^{j}x\\}}$

is called a _CO-measure_. Here _CO_ stands for _Closed-Orbit_ ; see for
instance Sigmund [S2]. If $(X,T)$ has the specification property, the set of
CO-measures is dense in the set of $T$-invariant Borel probability measures on
$X$ (see [S2, Th. 1]).

Let $(N_{k})_{k\geq 1}$ be a strictly increasing sequence of integers. We
denote by $C_{p,(N_{k})}$ the set of all $(p^{N_{k}}-1)$-th roots of $1$:

$C_{p,(N_{k})}=\bigl{\\{}\lambda\in\mathbb{T}\;;\;\lambda^{p^{N_{k}}-1}=1\quad\textrm{for
some}\ k\geq 1\bigr{\\}}.$

Let $D_{p,(N_{k})}$ be the family of CO-measures associated to elements
$\lambda$ of $C_{p,(N_{k})}$:

$D_{p,(N_{k})}=\bigl{\\{}\mu_{\lambda}\;;\;\lambda\in
C_{p,(N_{k})}\bigr{\\}}.$

We are now going to prove the following density result, which will be crucial
for the proof of Theorem 1.5:

######  Theorem 2.3.

The set $D_{p,(N_{k})}$ is dense in $(\mathcal{P}_{p}(\mathbb{T}),w^{*})$.

###### Proof.

Our aim is to show that given $\mu\in\mathcal{P}_{p}(\mathbb{T})$,
$f_{1},\dots,f_{l}$ belonging to $C(\mathbb{T})$, and $\varepsilon>0$, there
exists $\lambda\in C_{p,(N_{k})}$ such that

$\Bigl{|}\int_{\mathbb{T}}f_{i}\,d\mu_{\lambda}-\int_{\mathbb{T}}f_{i}\,d\mu\Bigr{|}<\varepsilon\quad\textrm{for
every}\ i\in\\{1,\ldots,l\\}.$

Since CO-measures are $w^{*}$-dense in $\mathcal{P}_{p}(\mathbb{T})$, we can
suppose without loss of generality that $\mu$ is a CO-measure, which we write
as

$\mu_{z}=\dfrac{1}{N}\sum_{j=0}^{N-1}\delta_{\\{z^{p^{j}}\\}}\quad\textrm{for
some}\ z\in\mathbb{T}\ \textrm{and}\ N\geq 1\ \textrm{such that}\
z^{p^{N}-1}=1.$

Because Lipschitz functions, with respect to the distance induced by
$\mathbb{C}$ on $\mathbb{T}$, are dense in $C(\mathbb{T})$ by the Stone-
Weierstrass theorem, we can also suppose without loss of generality that the
functions $f_{1},\dots,f_{l}$ are Lipschitz. Let $C>0$ be such that for every
$i\in\\{1,\ldots,l\\}$ and every $z_{1},z_{2}\in\mathbb{T}$,
$|f_{i}(z_{1})-f_{i}(z_{2})|\leq C\,|z_{1}-z_{2}|$. We are looking for
$\lambda\in\mathbb{T}$ and $k\geq 1$ with $\lambda^{p^{N_{k}}-1}=1$ such that

$\Bigl{|}\dfrac{1}{N}\sum_{j=0}^{N-1}f_{i}(z^{p^{j}})-\dfrac{1}{N_{k}}\sum_{j=0}^{N_{k}-1}f_{i}(\lambda^{p^{j}})\bigr{|}<\varepsilon\quad\textrm{for
every}\ i\in\\{1,\ldots,l\\}.$

Fix $\varepsilon^{\prime}>0$. Let $N_{\varepsilon^{\prime}}$ be given by the
specification property. Let $k\geq 1$ be such that
$N_{k}>2N_{\varepsilon}^{\prime}+2$. Applying Definition 2.2 to
$x_{1}=x_{2}=z$, $a_{1}=0$, $b_{1}=N_{k}-2N_{\varepsilon^{\prime}}-2$,
$a_{2}=b_{2}=b_{1}+N_{\varepsilon^{\prime}}+1$, and $d=N_{k}$, we obtain the
existence of $\lambda\in\mathbb{T}$ with $\lambda^{p^{N_{k}}}=\lambda$ such
that, for every $j=0,\ldots,N_{k}-2N_{\varepsilon^{\prime}}-2$,

$\bigl{|}z^{p^{j}}-\lambda^{p^{j}}\bigr{|}<\varepsilon^{\prime},\quad\textrm{
and hence }\quad\bigl{|}f_{i}(z^{p^{j}})-f_{i}(\lambda^{p^{j}})\bigr{|}\leq
C\varepsilon^{\prime}.$

Then, for every $i\in\\{1,\ldots,l\\}$,

$\Bigl{|}\dfrac{1}{N_{k}-2N_{\varepsilon^{\prime}}-1}\sum_{j=0}^{N_{k}-2N_{\varepsilon^{\prime}}-2}f_{i}(z^{p^{j}})-\dfrac{1}{N_{k}-2N_{\varepsilon^{\prime}}-1}\sum_{j=0}^{N_{k}-2N_{\varepsilon^{\prime}}-2}f_{i}(\lambda^{p^{j}})\Bigr{|}\leq
C\varepsilon^{\prime}.$

Now

$\displaystyle\Bigl{|}\dfrac{1}{N_{k}-2N_{\varepsilon^{\prime}}-1}\sum_{j=0}^{N_{k}-2N_{\varepsilon^{\prime}}-2}f_{i}(z^{p^{j}})$
$\displaystyle-\dfrac{1}{N_{k}}\sum_{j=0}^{N_{k}-1}f_{i}(z^{p^{j}})\Bigr{|}\leq\dfrac{1}{N_{k}}\sum_{j=N_{k}-2N_{\varepsilon^{\prime}}-1}^{N_{k}-1}|f_{i}(z^{p^{j}})|$
$\displaystyle+\Bigl{|}\dfrac{1}{N_{k}-2N_{\varepsilon^{\prime}}-1}-\dfrac{1}{N_{k}}\Bigr{|}\sum_{j=0}^{N_{k}-2N_{\varepsilon^{\prime}}-2}|f_{i}(z^{p^{j}})|$
$\displaystyle\leq\frac{2N_{\varepsilon^{\prime}}+1}{N_{k}}\,\bigl{|}\bigl{|}f_{i}\bigr{|}\bigr{|}_{\infty,\mathbb{T}}+\Bigl{(}1-\dfrac{N_{k}-2N_{\varepsilon^{\prime}}-1}{N_{k}}\Bigr{)}\,\bigl{|}\bigl{|}f_{i}\bigr{|}\bigr{|}_{\infty,\mathbb{T}}$
$\displaystyle\leq\dfrac{4N_{\varepsilon^{\prime}}+2}{N_{k}}\,\bigl{|}\bigl{|}f_{i}\bigr{|}\bigr{|}_{\infty,\mathbb{T}}\leq
6\,\dfrac{N_{\varepsilon^{\prime}}}{N_{k}}\,\bigl{|}\bigl{|}f_{i}\bigr{|}\bigr{|}_{\infty,\mathbb{T}}$

and

$\displaystyle\Bigl{|}\dfrac{1}{N_{k}-2N_{\varepsilon^{\prime}}-1}\sum_{j=0}^{N_{k}-2N_{\varepsilon^{\prime}}-2}f_{i}(\lambda^{p^{j}})$
$\displaystyle-\dfrac{1}{N_{k}}\sum_{j=0}^{N_{k}-1}f_{i}(\lambda^{p^{j}})\Bigr{|}\leq
6\,\dfrac{N_{\varepsilon^{\prime}}}{N_{k}}\,\bigl{|}\bigl{|}f_{i}\bigr{|}\bigr{|}_{\infty,\mathbb{T}}.$

Therefore

$\Bigl{|}\dfrac{1}{N_{k}}\sum_{j=0}^{N_{k}-1}f_{i}(z^{p^{j}})-\dfrac{1}{N_{k}}\sum_{j=0}^{N_{k}-1}f_{i}(\lambda^{p^{j}})\Bigr{|}\leq
C\varepsilon^{\prime}+12\,\dfrac{N_{\varepsilon^{\prime}}}{N_{k}}\,\bigl{|}\bigl{|}f_{i}\bigr{|}\bigr{|}_{\infty,\mathbb{T}}.$

Let now $r_{k}$ be the unique integer with $r_{k}N\leq N_{k}<(r_{k}+1)N$.
Then, proceeding in the same way as above, we obtain that

$\displaystyle\Bigl{|}\dfrac{1}{N_{k}}\sum_{j=0}^{N_{k}-1}f_{i}(z^{p^{j}})-\dfrac{1}{r_{k}.N}\sum_{j=0}^{r_{k}.N-1}f_{i}(z^{p^{j}})\Bigr{|}$
$\displaystyle\leq\dfrac{1}{N_{k}}\sum_{j=r_{k}N}^{N_{k}-1}|f_{i}(z^{p^{j}})|+\Bigl{|}\dfrac{1}{N_{k}}-\dfrac{1}{r_{k}.N}\Bigr{|}\sum_{j=0}^{r_{k}N-1}|f_{i}(z^{p^{j}})|$
$\displaystyle\leq
2\,\dfrac{N}{N_{k}}\,\bigl{|}\bigl{|}f_{i}\bigr{|}\bigr{|}_{\infty,\mathbb{T}}.$

Since $N$ is fixed and
$\textstyle{N_{k}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{+\infty}$,
we can choose $N_{k}>2N_{\varepsilon^{\prime}}+2$ sufficiently large so that
$\max(12\,\frac{N_{\varepsilon^{\prime}}}{N_{k}},2\,\frac{N}{N_{k}})\,\bigl{|}\bigl{|}f_{i}\bigr{|}\bigr{|}_{\infty,\mathbb{T}}<\varepsilon^{\prime}$
for every $i\in\\{1\ldots l\\}$. As

$\dfrac{1}{r_{k}.N}\sum_{j=0}^{r_{k}N-1}f_{i}(z^{p^{j}})=\dfrac{1}{N}\sum_{j=0}^{N-1}f_{i}(z^{p^{j}}),$

we get that

$\Bigl{|}\dfrac{1}{N}\sum_{j=0}^{N-1}f_{i}(z^{p^{j}})-\dfrac{1}{N_{k}}\sum_{j=0}^{N_{k}-1}f_{i}(\lambda^{p^{j}})\Bigr{|}\leq(C+2)\varepsilon^{\prime}.$

Taking $\varepsilon^{\prime}$ so small that
$(C+2)\varepsilon^{\prime}<\varepsilon$ yields the result we are looking for.
∎

######  Remark 2.4.

The argument presented above actually holds in a much more general setting,
and shows the following result. Let $(X,T)$ be a dynamical system with the
periodic specification property. Given a strictly increasing sequence
$(N_{k})_{k\geq 1}$ of integers, let

$C_{T,(N_{k})}=\bigl{\\{}x\in X\;;\;T^{N_{k}}x=x\quad\textrm{for some}\ k\geq
1\bigr{\\}}$

denote the set of periodic points for $T$ having a period within the set
$\\{N_{k}\;;\;k\geq 1\\}$. Let $D_{T,(N_{k})}=\bigl{\\{}\mu_{x}\;;\;x\in
C_{T,(N_{k})}\bigr{\\}}$. Then $D_{T,(N_{k})}$ is dense in the set
$\mathcal{P}_{T}(X)$ of $T$-invariant Borel probability measures on $X$,
endowed with the $w^{*}$-topology.

###  2.b. Proof of Theorem 2.1

Let $(c_{n})_{n\geq 0}$ be a sequence of integers satisfying assumption (H),
and let $t_{1},\dots,t_{r}$, $h_{1},\dots,h_{d}$ and $I\subseteq\mathbb{N}$ be
given by (H). For any $0<\gamma<1$, consider the set

$\displaystyle G_{p,(c_{n})}^{\,\gamma}=\bigl{\\{}\mu\in$
$\displaystyle\mathcal{P}_{p}(\mathbb{T})\;;\;\forall\,j\in\\{1,\ldots,r\\}\;\;\widehat{\mu}(t_{j})\neq
0\quad\textrm{and}$ $\displaystyle\forall\,n_{0},\ \exists\,n\geq n_{0},\
\exists\,l\in\\{1,\ldots,d\\}\;;\;|\widehat{\mu}(h_{l}c_{n})|>\gamma\min_{1\leq
j\leq r}|\widehat{\mu}(t_{j})|\bigr{\\}}.$

The interest of introducing this somewhat strange-looking set is the following
fact.

######  Fact 2.5.

If $\mu$ belongs to $G_{p,(c_{n})}^{\,\gamma}$, then there exists
$l\in\\{1\ldots d\\}$ such that

$\limsup\limits_{n\to+\infty}|\widehat{\mu}(h_{l}c_{n})|>0.$

In particular, $T_{c_{n}}\mu\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
3.0pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
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0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
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3.0pt}}}}}}\ignorespaces{\hbox{\kern 23.0pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces{\hbox{\kern
11.05365pt\raise-0.1pt\hbox{\hbox{\kern
0.0pt\raise-1.75pt\hbox{$\textstyle{\scriptstyle/}$}}}}}{\hbox{\kern
23.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}\ignorespaces}}}}\ignorespaces\textrm{Leb}$ as
${n}\longrightarrow{+\infty.}$

###### Proof.

Let $\mu\in G_{p,(c_{n})}^{\,\gamma}$. There exists $l\in\\{1\ldots d\\}$ such
that $|\widehat{\mu}(h_{l}c_{n})|>\gamma\min_{1\leq j\leq
r}|\widehat{\mu}(t_{j})|$ for infinitely many $n$’s, and so

$\limsup\limits_{n\to+\infty}|\widehat{\mu}(h_{l}c_{n})|\geq\gamma\min_{1\leq
j\leq r}|\widehat{\mu}(t_{j})|>0.$

Hence $\widehat{T_{c_{n}}\mu}(h_{l}){\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
3.0pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
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23.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces{\hbox{\kern
11.05365pt\raise-0.1pt\hbox{\hbox{\kern
0.0pt\raise-1.75pt\hbox{$\textstyle{\scriptstyle/}$}}}}}{\hbox{\kern
23.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}\ignorespaces}}}}\ignorespaces}0$ as
${n}\longrightarrow{+\infty}$, and as $h_{l}\neq 0$ this implies that
$T_{c_{n}}\mu\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
3.0pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
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0.0pt\hbox{$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
6.40012pt\raise 4.97223pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.97223pt\hbox{$\scriptstyle{w^{*}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 23.0pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces{\hbox{\kern
11.05365pt\raise-0.1pt\hbox{\hbox{\kern
0.0pt\raise-1.75pt\hbox{$\textstyle{\scriptstyle/}$}}}}}{\hbox{\kern
23.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}\ignorespaces}}}}\ignorespaces\textrm{Leb}$. ∎

We first prove:

######  Lemma 2.6.

For every $0<\gamma<1$, the set $G_{p,(c_{n})}^{\,\gamma}$ is a dense
$G_{\delta}$ subset of $(\mathcal{P}_{p}(\mathbb{T}),w^{*})$.

###### Proof.

The set $G_{p,(c_{n})}^{\,\gamma}$ is clearly $G_{\delta}$ in
$(\mathcal{P}_{p}(\mathbb{T}),w^{*})$, so we only need to show that it is
dense. Order the infinite set $I\subseteq\mathbb{N}$ as a strictly increasing
sequence $(N_{k})_{k\geq 1}$.

We have the following:

######  Fact 2.7.

Let $\mu$ belong to $D_{p,(N_{k})}$. There exist $i\in\\{1,\ldots,r\\}$ and
$l\in\\{1,\ldots,d\\}$ such that
$\widehat{\mu}{(h_{l}c_{n})}=\widehat{\mu}(t_{i})$ for infinitely many
integers $n$.

###### Proof of Fact 2.7.

There exists $\lambda\in C_{p,(N_{k})}$, with $\lambda^{p^{N_{k}}-1}=1$ for
some $k\geq 1$, such that

$\mu=\mu_{\lambda}=\dfrac{1}{N_{k}}\sum_{j=0}^{N_{k}-1}\delta_{\\{\lambda^{p^{j}}\\}}.$

For every $a\in\mathbb{Z}$, we have

$\widehat{\mu}(a)=\dfrac{1}{N_{k}}\sum_{j=0}^{N_{k}-1}\lambda^{a.p^{j}}.$

Since $N_{k}\in I$, there exist $i\in\\{1,\dots,r\\}$ and
$l\in\\{1,\dots,d\\}$ such that $h_{l}c_{n}\equiv t_{i}\mod(p^{N_{k}}-1)$ for
infinitely many $n$’s. Hence

$h_{l}c_{n}p^{j}\equiv t_{i}p^{j}\mod(p^{N_{k}}-1)\quad\textrm{for every
}0\leq j<N_{k}.$

As $\lambda^{p^{N_{k}}-1}=1$, it follows that
$\lambda^{h_{l}c_{n}p^{j}}=\lambda^{t_{i}p^{j}}$. This yields that
$\widehat{\mu}(h_{l}c_{n})=\widehat{\mu}(t_{i})$. ∎

Let now $\mathcal{V}$ be a non-empty open subset of
$(\mathcal{P}_{p}(\mathbb{T}),w^{*})$. By Theorem 2.3, there exists $\mu\in
D_{p,(N_{k})}\cap\mathcal{V}$. Let $i\in\\{1,\dots,r\\}$ and
$l\in\\{1,\dots,d\\}$ be such that
$\widehat{\mu}(h_{l}c_{n})=\widehat{\mu}(t_{i})$ for every $n$ belonging to a
certain infinite subset $D$ of $\mathbb{N}$. Then

$\limsup_{n\to+\infty}|\widehat{\mu}(h_{l}c_{n})|\geq|\widehat{\mu}(t_{i})|\geq\min_{1\leq
j\leq r}|\widehat{\mu}(t_{j})|,$

and if $\widehat{\mu}(t_{j})\neq 0$ for every $j\in\\{1,\ldots,r\\}$, then
$\mu$ belongs to $G_{p,(c_{n})}^{\gamma}$. Hence
$G_{p,(c_{n})}^{\gamma}\cap\mathcal{V}\neq\emptyset$ in this case.

Suppose now that $\min\limits_{1\leq j\leq r}|\widehat{\mu}(t_{j})|=0$, and
write $\\{1,\dots,r\\}=I\,\cup\,J$, where

$I=\bigl{\\{}j\in\\{1,\ldots,r\\}\;;\;\widehat{\mu}(t_{j})=0\bigr{\\}}\quad\textrm{and}\quad
J=\bigl{\\{}j\in\\{1,\ldots,r\\}\;;\;\widehat{\mu}(t_{j})\neq 0\bigr{\\}}.$

For any $0<\rho<1$, consider the measure
$\mu_{\rho}=\rho\delta_{1}+(1-\rho)\mu$: it is $T_{p}$-invariant and belongs
to $\mathcal{V}$ if $\rho$ is sufficiently small. For every
$j\in\\{1,\ldots,r\\}$,
$\widehat{\mu}_{\rho}(t_{j})=\rho+(1-\rho)\widehat{\mu}(t_{j})$, so that
$\widehat{\mu}_{\rho}(t_{j})=\rho>0$ for every $j\in I$. If $\rho$ is
sufficiently small, $|\widehat{\mu}_{\rho}(t_{j})|>0$ for every $j\in J$ and
thus $\min\limits_{1\leq j\leq r}|\widehat{\mu}_{\rho}(t_{j})|>0$. Since
$\widehat{\mu}_{\rho}(h_{l}c_{n})=\rho+(1-\rho)\widehat{\mu}(t_{i})=\widehat{\mu}_{\rho}(t_{i})$
for every $n\in D,$ it follows that $\widehat{\mu}_{\rho}$ belongs to
$G_{p,(c_{n})}^{\gamma}\cap\mathcal{V}$ in this case as well. Lemma 2.6 is
proved. ∎

The last step in our proof of Theorem 2.1 is the following classical fact:

######  Fact 2.8.

The set $\mathcal{P}_{p,c}(\mathbb{T})$ is a dense $G_{\delta}$ subset of
$(\mathcal{P}_{p}(\mathbb{T}),w^{*})$.

###### Proof.

It is a known result that if $X$ is a Polish space, the set
$\mathcal{P}_{c}(X)$ of continuous probability measures on $X$ is a
$G_{\delta}$ subset of the set $\mathcal{P}(X)$ of all Borel probability
measures on $X$, endowed with the $w^{*}$-topology (see for instance [DGS,
Proposition 2.16] or [GM, Fact 3.2]). So $\mathcal{P}_{p,c}(\mathbb{T})$ is
$G_{\delta}$ in $(\mathcal{P}_{p}(\mathbb{T}),w^{*})$. The density of
$\mathcal{P}_{p,c}(\mathbb{T})$ in $\mathcal{P}_{p}(\mathbb{T})$ is proved in
[S1], using the density of CO-measures in $\mathcal{P}_{p}(\mathbb{T})$. It
can also be retrieved by using the following elementary observation: there
exists a sequence $(\mu_{k})_{k\geq 1}$ of elements of
$\mathcal{P}_{p,c}(\mathbb{T})$ such that
$\mu_{k}\xrightarrow{\text{$w^{*}$}}\delta_{1}$.

The measures $\mu_{k}$ can be constructed as Cantor-type measures, also called
$p$-Bernoulli in [L1], [L2], or [FS]: given a $p$-tuple
$\Theta=(\theta_{0},\theta_{1},\dots,\theta_{p-1})$ of elements of $(0,1)$
with $\sum_{j=0}^{p-1}\theta_{i}=1$, let $m_{\Theta}$ be the product measure

$m_{\Theta}=\bigotimes_{n\geq
1}\,\,\Bigl{(}\,\sum_{j=0}^{p-1}\theta_{j}\delta_{\\{j\\}}\,\Bigr{)}\quad\textrm{on}\quad\\{0,1,\dots,p-1\\}^{\mathbb{N}},$

and let $\mu_{\Theta}\in\mathcal{P}(\mathbb{T})$ be the image measure of
$m_{\Theta}$ by the map
$\textstyle{\Phi:\\{0,1,\dots,p-1\\}^{\mathbb{N}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbb{T}}$
defined by

$\Phi\bigl{(}(\omega_{n})_{n\geq 1}\bigr{)}=\exp\Bigl{(}2i\pi\sum_{n\geq
1}\omega_{n}p^{-n}\Bigr{)}.$

Each measure $\mu_{\Theta}$ is easily seen to belong to
$\mathcal{P}_{p,c}(\mathbb{T})$, and
${\mu_{\Theta}}\xrightarrow{\text{$w^{*}$}}{\delta_{1}}$ as
$\textstyle{\Theta\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(1,0,\dots,0).}$

Once we have obtained a sequence $(\mu_{k})_{k\geq 1}$ of elements of
$\mathcal{P}_{p,c}(\mathbb{T})$ such that
$\mu_{k}\xrightarrow{\text{$w^{*}$}}\delta_{1}$, the density of
$\mathcal{P}_{p,c}(\mathbb{T})$in $\mathcal{P}_{p}(\mathbb{T})$ immediately
follows, since for each measure $\mu\in\mathcal{P}_{p}(\mathbb{T})$,
$(\mu_{k}*\mu)_{k\geq 1}$ is a sequence of $T_{p}$-invariant continuous
measures which converges $w^{*}$ to $\mu$. ∎

###### Proof of Theorem 2.1.

It follows from Lemma 2.6 and Fact 2.8 and from the Baire Category Theorem
that for any $\gamma\in(0,1)$, the set
$G_{p,(c_{n})}^{\gamma}\cap\mathcal{P}_{p,c}(\mathbb{T})$ is residual in
$(\mathcal{P}_{p}(\mathbb{T}),w^{*})$. By Fact 2.5, the set

$\bigl{\\{}\mu\in\mathcal{P}_{p,c}(\mathbb{T})\;;\;\exists
l\in\\{1,\ldots,d\\}\;\;\limsup\limits_{n\to+\infty}|\widehat{\mu}(h_{l}c_{n})|>0\bigr{\\}}$

is residual in $(\mathcal{P}_{p}(\mathbb{T}),w^{*})$, and hence
$G_{p,(c_{n})}{}$ is residual as well. Theorem 2.1 is proved. ∎

##  3\. Proofs of Theorems 1.5 and 1.7, and further examples

In this section, we apply Theorem 2.1 to various classes of sequences
$(c_{n})_{n\geq 0}$, and show that generically in the Baire Category sense, a
measure $\mu\in\mathcal{P}_{p}(\mathbb{T})$ has infinitely many “large”
Fourier coefficients along the sequence $(c_{n})_{n\geq 0}$, or along some
dilated sequence $(a.c_{n})_{n\geq 0}$ for some
$a\in\mathbb{Z}\setminus\\{0\\}$. We begin by proving Theorem 1.5.

###  3.a. Disproving Conjecture (C3): proof of Theorem 1.5

Let $p\geq 2$. Given another integer $q\geq 2$ (not necessarily
multiplicatively independent from $p$), we consider the sequence
$c_{n}=q^{n}$, $n\geq 0$. In order to show that the sets

$S_{p,q}:=\bigl{\\{}\mu\in\mathcal{P}_{p,c}(\mathbb{T})\;;\;\limsup\limits_{n\to+\infty}|\widehat{\mu}(q^{n})|>0\bigr{\\}}$

and

$G_{p,q}{}=\\{\mu\in\mathcal{P}_{p,c}(\mathbb{T})\;;\;T_{q^{n}}\mu\lx@xy@svg{\hbox{\raise
0.0pt\hbox{\kern
3.0pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
6.40012pt\raise 4.97223pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.97223pt\hbox{$\scriptstyle{w^{*}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 23.0pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces{\hbox{\kern
11.05365pt\raise-0.1pt\hbox{\hbox{\kern
0.0pt\raise-1.75pt\hbox{$\textstyle{\scriptstyle/}$}}}}}{\hbox{\kern
23.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}\ignorespaces}}}}\ignorespaces\textrm{Leb}\\}$

are dense in $(\mathcal{P}_{p,c}(\mathbb{T}),w^{*})$ it suffices to show that
this sequence $(c_{n})_{n\geq 0}$ satisfies assumption (H), and then to apply
Theorem 2.1. That assumption (H) is satisfied is a consequence of the
following lemma, which relies on considerations from elementary number theory:

######  Lemma 3.1.

Let $p,q\geq 2$. There exists an integer $N_{0}\geq 1$ such that for every
$a\geq 1$ sufficiently large, the following assertion holds:

for every $N\in I:=N_{0}.\mathbb{N}+1$, there exists an integer $r_{N}\geq 1$
such that

$q^{a+k.r_{N}}\equiv q^{a}\mod(p^{N}-1)\quad\textrm{for every}\ k\geq 0.$

###### Proof.

Write $q$ as $q=q_{1}^{b_{1}}\\!\\!\dots q_{s}^{b_{s}}$, where
$q_{1},\dots,q_{s}$ are primes and $b_{1},\dots,b_{s}$ are positive integers.
Let $a_{0}\geq 1$ be such that $1<p<q_{i}^{a_{0}b_{i}}$ for every
$i\in\\{1,\ldots,s\\}$. A first step in the proof of Lemma 3.1 is to show the
following

######  Fact 3.2.

Let $u\geq 1$ be a positive integer. Let $\gamma\geq 1$ be such that for every
$i\in\\{1,\ldots,s\\}$ and every $v\in\\{1,\ldots,u\\}$, $q_{i}^{\gamma}$ does
not divide $p^{v}-1$. There exist integers $N_{1}>u,\dots,N_{s}>u$ such that
for every $i\in\\{1,\ldots,s\\}$ and every
$N\in\mathbb{N}\setminus\bigcup_{j=1}^{s}N_{j}.\mathbb{N}$,

$p^{N}\not\equiv 1\mod q_{i}^{\gamma}.$

###### Proof.

Let $i\in\\{1,\ldots,s\\}$. If $p^{N}\not\equiv 1\mod q_{i}^{\gamma}$ for
every $N\geq 1$, then clearly $p^{N}\not\equiv 1\mod q_{i}^{\gamma}$ for every
$N\in\mathbb{N}\setminus\bigcup_{j=1}^{s}N_{j}.\mathbb{N}$, whatever the
choice of the integers $N_{1},\dots,N_{s}$. So we can suppose without loss of
generality that there exists an integer $N\geq 1$ such that $p^{N}\equiv 1\mod
q_{i}^{\gamma}$. Let $N_{i}$ be the smallest such integer. Necessarily,
$N_{i}>u$, since else $q_{i}^{\gamma}$ would divide $p^{v}-1$ for some
$v\in\\{1,\ldots,u\\}$. Moreover any integer $N$ such that $p^{N}\equiv 1\mod
q_{i}^{\gamma}$ is a multiple of $N_{i}$. It follows that $p^{N}\not\equiv
1\mod q_{i}^{\gamma}$ for every $N\in\mathbb{N}\setminus N_{i}.\mathbb{N}$. ∎

We apply Fact 3.2 to $u=1$ and $\gamma=a_{0}.\max_{1\leq i\leq s}b_{i}$. Let
$N_{1},\ldots,N_{s}$ be given by Fact 3.2. Since $N_{j}\geq 2$ for every
$j\in\\{1,\ldots,s\\}$, the set
$J:=\mathbb{N}\setminus\bigcup_{j=1}^{s}N_{j}.\mathbb{N}$ is infinite. Set
$N_{0}=N_{1}\ldots N_{l}$. Then $I:=N_{0}.\mathbb{N}+1$ is contained in $J$.

Fix $N\in I$. For each $i\in\\{1,\ldots,s\\}$, let $0\leq a_{i,N}<\gamma$ be
the largest integer such that $q_{i}^{a_{i,N}}$ divides $p^{N}-1$, and write
$p^{N}-1=q_{i}^{a_{i,N}}s_{i,N}$ for some integer $s_{i,N}\geq 1$ with
$\textrm{gcd}(s_{i,N},q_{i})=1$. By the Fermat-Euler Theorem, there exists
$r_{i,N}\geq 1$ such that $q_{i}^{r_{i,N}}\equiv 1\mod s_{i,N}$. Set
$r_{N}=r_{1,N}.r_{2,N}\dots r_{s,N}$. Then for every $l\geq 1$ and every
$i\in\\{1,\ldots,s\\}$, $q_{i}^{l.r_{N}}\equiv 1\mod s_{i,N}$, so that
$q_{i}^{a_{i,N}+l.r_{N}}\equiv q_{i}^{a_{i,N}}\mod(p^{N}-1)$ for every
$i\in\\{1,\ldots,s\\}$. If $a$ is sufficiently large, we have
$a_{i,N}<\gamma<a.b_{i}$ for every $i\in\\{1,\ldots,s\\}$, so that
$q_{i}^{a.b_{i}+l.r_{N}}\equiv q_{i}^{a.b_{i}}\mod(p^{N}-1)$. Applying this to
$l=k.b_{i}$, $k\geq 1$, yields that
$(q_{i}^{b_{i}})^{a+k.r_{N}}\equiv(q_{i}^{b_{i}})^{a}\mod(p^{N}-1)$ for every
$i\in\\{1,\ldots,s\\}$, i.e. that $q^{a+k.r_{N}}\equiv q^{a}\mod(p^{N}-1)$. ∎

###### Proof of Theorem 1.5.

By Lemma 3.1 above, the sequence $(q^{n})_{n\geq 1}$ satisfies assumption (H).
The proof of Theorem 2.1 combined with Lemma 3.1 shows the density of

$S_{p,q}:=\bigl{\\{}\mu\in\mathcal{P}_{p,c}(\mathbb{T})\;;\;\limsup\limits_{n\to+\infty}|\widehat{\mu}(q^{n})|>0\bigr{\\}}$

in $(\mathcal{P}_{p,c}(\mathbb{T}),w^{*})$. ∎

######  Remark 3.3.

The proof of Theorem 1.5 does not make use of all the information provided by
Lemma 3.1: we apply it with $u=1$, the particular form of the set $I$ is not
used, and we only need the fact that for every $N\in I$, there exist
infinitely many $n$’s such that $q^{n}\equiv q^{a}\mod(p^{N}-1)$. This
additional information will be important, however, in the forthcoming proofs
of Theorems 3.4 and 1.7.

###  3.b. A generalisation of Theorem 1.5

In this section, we consider sequences $(c_{n})_{n\geq 0}$ of the following
form: $c_{n}=f_{1}(n)q_{1}^{n}+f_{2}(n)q_{2}^{n}+\cdots+f_{d}(n)q_{d}^{n}$,
$n\geq 0$, where for each $l\in\\{1,\ldots,d\\}$, $q_{l}\geq 2$ is an integer
and $f_{l}$ is a polynomial with coefficients in $\mathbb{Z}$. This class of
sequences is considered in [M] and [LMP], where the following result is
proved: if $p\geq 2$ admits a prime factor $p^{*}$ which does not divide one
of the integers $q_{i}$, $1\leq i\leq d$, then any measure
$\mu\in\mathcal{P}_{p}(\mathbb{T})$ which is ergodic and of positive entropy
is $(c_{n})$-generic. It follows that the set

$G^{\prime}_{p,(c_{n})}=\bigl{\\{}\mu\in\mathcal{P}_{p,c}(\mathbb{T})\;;\;T_{c_{n}}\mu\xrightarrow{\text{$w^{*}$}}\textrm{Leb}\
\textrm{on a set of upper density}\ 1\bigr{\\}}$

is residual in $(\mathcal{P}_{p}(\mathbb{T}),w^{*})$ – see Section 5.a for
details on this argument.

We complement this result by showing the following

######  Theorem 3.4.

If $(c_{n})_{n\geq 0}$ is a sequence of the form
$c_{n}=f_{1}(n)q_{1}^{n}+f_{2}(n)q_{2}^{n}+\cdots+f_{d}(n)q_{d}^{n}$, $n\geq
0$, where for each $l\in\\{1,\ldots,d\\}$, $q_{l}\geq 2$ is an integer and
$f_{l}$ is a polynomial with coefficients in $\mathbb{Z}$, then the set

$G_{p,(c_{n})}=\bigl{\\{}\mu\in\mathcal{P}_{p,c}(\mathbb{T})\;;\;T_{c_{n}}\mu\lx@xy@svg{\hbox{\raise
0.0pt\hbox{\kern
3.0pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
6.40012pt\raise 4.97223pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.97223pt\hbox{$\scriptstyle{w^{*}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 23.0pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces{\hbox{\kern
11.05365pt\raise-0.1pt\hbox{\hbox{\kern
0.0pt\raise-1.75pt\hbox{$\textstyle{\scriptstyle/}$}}}}}{\hbox{\kern
23.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}\ignorespaces}}}}\ignorespaces\emph{{Leb}}\
\textrm{as}\ \lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
6.00117pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{n\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
26.00117pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
26.00117pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{+\infty\bigr{\\}}}$}}}}}\ignorespaces}}}}\ignorespaces$

is residual in $(\mathcal{P}_{p}(\mathbb{T}),w^{*})$.

###### Proof.

Let us show that $(c_{n})_{n\geq 0}$ satisfies assumption (H). By Lemma 3.1,
there exist integers $a\geq 1$ and $N_{l}$, $1\leq l\leq d$, such that for
every $l\in\\{1,\ldots,d\\}$ and every $N\in I_{l}:=N_{l}.\mathbb{N}+1$, there
exists an integer $r_{l,N}\geq 1$ such that for every $k\geq 0$,

$q_{l}^{a+k.r_{l,N}}\equiv q_{l}^{a}\mod(p^{N}-1).$

The set $I=\bigcap_{l=1}^{d}I_{l}$ is infinite. If we set, for each $N\in I$,
$r_{N}=r_{1,N}\dots r_{d,N}$, we get that for every $N\in I$ and every $k\geq
0$,

$q_{l}^{a+k.r_{N}}\equiv q_{l}^{a}\mod(p^{N}-1)\quad\textrm{for every}\
l\in\\{1,\ldots,d\\}.$ (1)

For each $l\in\\{1,\ldots,d\\}$, write the polynomial $f_{l}$ as

$f_{l}(x)=\sum_{j=0}^{\Delta_{l}}b_{j}^{(l)}x^{j},\quad\textrm{where}\
b_{j}^{(l)}\in\mathbb{Z}\ \textrm{for every}\ j\in\\{0,\ldots,\Delta_{l}\\}.$

For every $N\in I$ and every integer $k^{\prime}\geq 0$, we have

$\displaystyle f_{l}\bigl{(}a+k^{\prime}(p^{N}-1)r_{N}\bigr{)}$
$\displaystyle=\sum_{j=0}^{\Delta_{l}}b_{j}^{(l)}\bigl{(}a+k^{\prime}(p^{N}-1)r_{N}\bigr{)}^{j}$
and $\displaystyle\bigl{(}a+k^{\prime}(p^{N}-1)r_{N}\bigr{)}^{j}$
$\displaystyle\equiv a^{j}\mod(p^{N}-1)\quad\textrm{for every}\ j\geq 0.$
Hence $\displaystyle f_{l}\bigl{(}a+k^{\prime}(p^{N}-1)r_{N}\bigr{)}$
$\displaystyle\equiv f_{l}(a)\mod(p^{N}-1)\quad\textrm{for every}\
l\in\\{1,\ldots,d\\}.$ (2) Putting together (1) and (2) yields that for every
$N\in I$, $\displaystyle c_{a+k^{\prime}(p^{N}-1)r_{,N}}$ $\displaystyle\equiv
c_{a}\mod(p^{N}-1)\quad\textrm{for every}\ k^{\prime}\geq 0,$

which implies that assumption (H) is true. Theorem 3.4 thus follows from
Theorem 2.1. ∎

######  Remark 3.5.

We notice that if $c_{n}=f(n)$ for some polynomial $f\in\mathbb{Z}[X]$, then
the set

$\bigl{\\{}\mu\in\mathcal{P}_{p,c}(\mathbb{T})\;;\;T_{c_{n}}\mu\lx@xy@svg{\hbox{\raise
0.0pt\hbox{\kern
3.0pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
6.40012pt\raise 4.97223pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.97223pt\hbox{$\scriptstyle{w^{*}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 23.0pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces{\hbox{\kern
11.05365pt\raise-0.1pt\hbox{\hbox{\kern
0.0pt\raise-1.75pt\hbox{$\textstyle{\scriptstyle/}$}}}}}{\hbox{\kern
23.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}\ignorespaces}}}}\ignorespaces\textrm{Leb}\bigr{\\}}$

is also residual in $(\mathcal{P}_{p}(\mathbb{T}),w^{*})$. Remark that in this
case, the sequence $(c_{n}x)_{n\geq 0}$ is uniformly distributed mod 1 for
every $x\in\mathbb{R}\setminus\mathbb{Q}$, and hence

$\dfrac{1}{N}\sum_{n=0}^{N-1}\exp(2i\pi hc_{n}x)\lx@xy@svg{\hbox{\raise
0.0pt\hbox{\kern
3.0pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
23.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
23.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{0}$}}}}}\ignorespaces}}}}\ignorespaces\quad\textrm{as}\
\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
7.56248pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{N\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
27.56248pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
27.56248pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{+\infty}$}}}}}\ignorespaces}}}}\ignorespaces\
\textrm{for every}\ h\in\mathbb{Z}\setminus\\{0\\}.$ (3)

If $\mu$ belongs to $\mathcal{P}_{p,c}(\mathbb{T})$, integrating (3) with
respect to the measure $\mu$ yields that

$\displaystyle\dfrac{1}{N}\sum_{n=0}^{N-1}\widehat{\mu}(hc_{n})$
$\displaystyle\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
3.0pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
23.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
23.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{0}$}}}}}\ignorespaces}}}}\ignorespaces\quad\textrm{for
every}\
h\in\mathbb{Z}\setminus\\{0\\},\quad\textrm{i.e.}\quad\dfrac{1}{N}\sum_{n=0}^{N-1}T_{c_{n}}\mu\xrightarrow{\text{$w^{*}$}}\textrm{Leb}.$

###  3.c. Further examples: proof of Theorem 1.7

Let $(c_{n})_{n\geq 0}$ be defined by a linear recursion: there exist $L\geq
1$ and coefficients $a_{1},\dots,a_{L}$ in $\mathbb{Z}$ with $a_{L}\neq 0$
such that

$c_{n}=a_{1}c_{n-1}+a_{2}c_{n-2}+\cdots+a_{L}c_{n-L}\quad\textrm{for every}\
n\geq L.$

Let $p\geq 2$. Meiri introduces in [M] the following two assumptions on the
sequence $(c_{n})_{n\geq 0}$:

  1. (a)

$(c_{n})_{n\geq 0}$ has no non-constant arithmetic subsequence;

  2. (b)

$a_{L}$ and $p$ are relatively prime.

It is observed in [M, Prop. 5.1] that assumption (a) is satisfied as soon as
the following property holds:

  1. (a’)

if $\lambda_{1},\dots,\lambda_{L^{\prime}}$, with $1\leq L^{\prime}\leq L$,
are the distinct roots of the recursion polynomial
$p(x)=x^{L}-\sum_{l=1}^{L}a_{l}x^{L-l}$, then none of the quantities
$\lambda_{i}$ and $\lambda_{i}/\lambda_{j}$, $1\leq i<j\leq L^{\prime}$ is a
root of unity.

If assumptions (a) and (b) are satisfied, any measure
$\mu\in\mathcal{P}_{p}(\mathbb{T})$ which is ergodic and of positive entropy
is such that $T_{c_{n}}\mu\xrightarrow{\text{$w^{*}$}}{}\textrm{Leb}$ as
$\textstyle{n\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{+\infty}$
(this is a consequence of [M, Th. 5.2]). In particular, the set

$G^{\prime}_{p,(c_{n})}=\bigl{\\{}\mu\in\mathcal{P}_{p,c}(\mathbb{T})\;;\;T_{c_{n}}\mu\xrightarrow{\text{$w^{*}$}}\textrm{Leb}\quad\textrm{along
a set of upper density}\ 1\bigr{\\}}$

is residual in $(\mathcal{P}_{p}(\mathbb{T}),w^{*})$. See also [LMP, Th. 4.3].

Theorem 1.7 complements these results by showing that under the sole
assumption (b), the set

$G_{p,(c_{n})}=\bigl{\\{}\mu\in\mathcal{P}_{p,c}(\mathbb{T})\;;\;T_{c_{n}}\mu\lx@xy@svg{\hbox{\raise
0.0pt\hbox{\kern
3.0pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
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0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.97223pt\hbox{$\scriptstyle{w^{*}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 23.0pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces{\hbox{\kern
11.05365pt\raise-0.1pt\hbox{\hbox{\kern
0.0pt\raise-1.75pt\hbox{$\textstyle{\scriptstyle/}$}}}}}{\hbox{\kern
23.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}\ignorespaces}}}}\ignorespaces\textrm{Leb}\bigr{\\}}$

is residual in $(\mathcal{P}_{p}(\mathbb{T}),w^{*})$.

###### Proof of Theorem 1.7.

Consider the matrix $A\in M_{L}(\mathbb{Z})$ given by

$A=\begin{pmatrix}a_{1}&a_{2}&\dots&\dots&a_{L}\\\ 1&0&\dots&\dots&0\\\
0&1&\ddots&&0\\\ \vdots&\ddots&\ddots&\ddots&\vdots\\\
0&\dots&0&1&0\end{pmatrix}.$

Setting
$C_{n}={}^{\textrm{T}}\\!\begin{pmatrix}c_{n}&c_{n-1}&\dots&c_{n-L+1}\end{pmatrix}$
for every $n\geq L-1$, we have $C_{n+1}=AC_{n}$ for every $n\geq L-1$. Since
$\det A=(-1)^{L+1}a_{L}$ and $a_{L}\neq 0$, $A$ is invertible as a matrix of
$M_{L}(\mathbb{Q})$, and

$A^{-1}=\dfrac{(-1)^{L+1}}{a_{L}}\,\textrm{adj}\,(A)$

where $\textrm{adj}\,(A)$, the adjoint (or adjugate) of $A$, is the transpose
of the matrix of the cofactors of $A$. Observe that $\textrm{adj}\,(A)\in
M_{L}(\mathbb{Z})$.

Let us decompose the integer $a_{L}$ as $a_{L}=q_{1}^{b_{1}}\dots
q_{s}^{b_{s}}$, where $q_{i}$ is prime and $b_{i}\geq 1$ for every
$i\in\\{1,\ldots,s\\}$. By Fact 3.2, there exist $\gamma\geq 1$ and an
infinite subset $I$ of $\mathbb{N}$ such that for every $N\in I$,
$p^{N}\not\equiv 1\mod q_{i}^{\gamma}$ for every $i\in\\{1,\ldots,s\\}$.
Hence, for every $N\in I$, $p^{N}-1$ can be written as
$p^{N}-1=q_{1}^{\beta_{1,N}}\\!\\!\\!\dots\,q_{s}^{\beta_{s,N}}r_{N}$, where
$0\leq\beta_{i,N}<\gamma$ and $\textrm{gcd}(q_{i},r_{N})=1$ for each
$i\in\\{1,\ldots,s\\}$. Since the prime factors of $a_{L}$ are exactly the
$q_{i}$’s, it follows that $a_{L}$ and $r_{N}$ are relatively prime, and hence
that $a_{L}$ is invertible modulo $r_{N}$: there exists an integer $d_{N}$
with $0\leq d_{N}<r_{N}$ such that $a_{L}.d_{N}\equiv 1\mod r_{N}$. Setting
$B_{N}=(-1)^{L+1}d_{N}\,\textrm{adj}(A)$, we observe that $B_{N}\in
M_{L}(\mathbb{Z})$ and that $AB_{N}\equiv B_{N}A\equiv I\mod r_{N}$. So $A$ is
invertible modulo $r_{N}$, and its inverse is $B_{N}$.

Consider now the set of matrices in $M_{L}(\mathbb{Z})$ consisting of all
powers $A^{n}$, $n\geq 0$, of $A$, taken modulo $r_{N}$. This set being
finite, there exist two integers $0\leq n_{1,N}<n_{2,N}$ such that
$A^{n_{1,N}}\equiv A^{n_{2,N}}\mod r_{N}$. Setting $n_{N}=n_{2,N}-n_{1,N}$,
$A^{n_{N}}\equiv I\mod r_{N}$, and thus the sequence $(C_{n})$ taken modulo
$r_{N}$ is periodic, with period $n_{N}$. It follows that the sequence
$(c_{n})$ itself taken modulo $r_{N}$ is periodic of period $n_{N}$, so that,
in particular, $c_{jn_{N}}\equiv c_{0}\mod r_{N}$ for every $j\geq 0$. Setting
$h_{N}=q_{1}^{\beta_{1,N}}\\!\\!\\!\dots\,q_{s}^{\beta_{s,N}}$ and remembering
that $p^{N}-1=h_{N}.r_{N}$, we obtain that $h_{N}.c_{jn_{N}}\equiv
h_{N}c_{0}\mod(p^{N}-1)$ for every $j\geq 0$. Since $0\leq\beta_{i,N}<\gamma$
for every $i\in\\{1,\ldots,s\\}$, the set $\\{h_{N}\;;\;N\in I\\}$ is finite
and consists of non-zero integers. Assumption (H) is satisfied, and the proof
is concluded as usual thanks to Theorem 2.1.∎

##  4\. The multidimensional case: proof of Theorem 1.8

In this section, $d\geq 2$ is an integer, and $A,B\in M_{d}(\mathbb{Z})$ are
two $d\times d$ matrices with integer coefficients such that $\det A\neq 0$
and $\det B\neq 0$. The matrix $A$ is supposed to be similar in
$M_{d}(\mathbb{C})$ to a diagonal matrix $D$ whose diagonal coefficients
$\lambda_{1},\dots,\lambda_{d}$ are not of modulus $1$. Let $P\in
GL_{d}(\mathbb{C})$ be such that $A=PDP^{-1}$. The matrix $B$ is supposed to
be invertible in $M_{d}(\mathbb{C})$, i.e. $\det B\neq 0$. Theorem 1.8 states
that the set

$G_{A,B}{}=\bigl{\\{}\mu\in\mathcal{P}_{A,c}(\mathbb{T}^{d})\;;\;T_{B^{n}}\mu\lx@xy@svg{\hbox{\raise
0.0pt\hbox{\kern
3.0pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 3.0pt\raise
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6.40012pt\raise 4.97223pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.97223pt\hbox{$\scriptstyle{w^{*}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 23.0pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces{\hbox{\kern
11.05365pt\raise-0.1pt\hbox{\hbox{\kern
0.0pt\raise-1.75pt\hbox{$\textstyle{\scriptstyle/}$}}}}}{\hbox{\kern
23.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}\ignorespaces}}}}\ignorespaces\textrm{Leb}\bigr{\\}}$

is residual in $(\mathcal{P}_{A}(\mathbb{T}^{d}),w^{*})$. The proof of Theorem
1.8 follows the same structure as those of Theorems 2.1 and 1.5, but certain
technical difficulties that come with the multidimensional setting must be
overcome.

We begin by proving an analogue of Theorem 2.3.

###  4.a. Some dense classes of discrete measures in
$\mathcal{P}_{A}(\mathbb{T}^{d})$

Let $(N_{k})_{k\geq 1}$ be a strictly increasing sequence of integers.
Consider the set

$\boldsymbol{C}_{A,(N_{k})}=\bigl{\\{}\boldsymbol{x}\in\mathbb{T}^{d}\;;\;A^{N_{k}}\boldsymbol{x}=\boldsymbol{x}\quad\textrm{for
some}\ k\geq 1\bigr{\\}}$

which consists of periodic points for $T_{A}$ having a period within the set
$\\{N_{k}\;;\;k\geq 1\\}$. For each
$\boldsymbol{x}\in\boldsymbol{C}_{A,(N_{k})}$, let $\mu_{\boldsymbol{x}}$ be
the measure defined by

$\mu_{\boldsymbol{x}}=\dfrac{1}{N_{k}}\sum_{j=0}^{N_{k}-1}\delta_{\\{A^{j}\boldsymbol{x}\\}}.$

It is a discrete $T_{A}$-invariant probability measure on $\mathbb{T}^{d}$.
Set

$\boldsymbol{D}_{A,(N_{k})}=\bigl{\\{}\mu_{\boldsymbol{x}}\;;\;\boldsymbol{x}\in\boldsymbol{C}_{A,(N_{k})}\bigr{\\}}.$

Taking inspiration from Theorem 2.3, we would like to show that the set
$\boldsymbol{D}_{A,(N_{k})}$ is dense in
$(\mathcal{P}_{A}(\mathbb{T}^{d}),w^{*})$. If $(\mathbb{T}^{d},T_{A})$ has the
periodic specification property, this is an immediate consequence of Remark
2.4. However, $T_{A}$ is known to have the periodic specification property
only in the case where $A$ is an hyperbolic automorphism of $\mathbb{T}^{d}$,
i.e. $\det A=\pm 1$ and $A$ has no eigenvalue of modulus $1$. Since $A$ is not
assumed here to be an automorphism of $\mathbb{T}^{d}$, we need to take a
different route. It will lead to the following weaker result, which is
fortunately sufficient for our purposes:

######  Theorem 4.1.

The convex hull of the set $\boldsymbol{D}_{A,(N_{k})}$ is dense in
$(\mathcal{P}_{A}(\mathbb{T}^{d}),w^{*})$.

###### Proof.

Denote by $\boldsymbol{F}_{A,(N_{k})}$ the $w^{*}$-closure in
$\mathcal{P}_{A}(\mathbb{T}^{d})$ of the convex hull of
$\boldsymbol{D}_{A,(N_{k})}$. This is a $w^{*}$-closed convex subset of
$\mathcal{P}_{A}(\mathbb{T}^{d})$, and also of the Banach space
$\mathcal{M}(\mathbb{T}^{d})$ of complex measures on $\mathbb{T}^{d}$, endowed
with the norm $||\mu||:=|\mu|(\mathbb{T}^{d})$. This space
$\mathcal{M}(\mathbb{T}^{d})$ is the dual space of
$(C(\mathbb{T}^{d}),||\,.\,||_{\infty,\mathbb{T}^{d}})$, the space of
continuous functions on $\mathbb{T}^{d}$.

Our aim is to show that
$\boldsymbol{F}_{A,(N_{k})}=\mathcal{P}_{A}(\mathbb{T}^{d})$. Suppose that it
is not the case, and that there exists a measure $\mu_{0}$ belonging to
$\mathcal{P}_{A}(\mathbb{T}^{d})\setminus\boldsymbol{F}_{A,(N_{k})}$.

Applying the Hahn-Banach Theorem in the locally convex space
$(\mathcal{M}(\mathbb{T}^{d}),w^{*})$, we obtain that there exists a
$w^{*}$-continuous linear functional
$L:\mathcal{M}(\mathbb{T}^{d})\longrightarrow\mathbb{C}$, as well as real
numbers $\gamma_{1}<\gamma_{2}$ such that

$\Re e(L(\mu))\leq\gamma_{1}<\gamma_{2}\leq\Re e(L(\mu_{0}))$

for every $\mu\in\boldsymbol{F}_{A,(N_{k})}$. Since any $w^{*}$-continuous
functional on $\mathcal{M}(\mathbb{T}^{d})=C(\mathbb{T}^{d})^{*}$ acts as
integration against an element of $C(\mathbb{T}^{d})$, there exists a function
$f\in C(\mathbb{T}^{d})$ such that

$\Re e\int_{\mathbb{T}}f\,d\mu\leq\gamma_{1}<\gamma_{2}\leq\Re
e\int_{\mathbb{T}^{d}}f\,d\mu_{0}$

for every $\mu\in\boldsymbol{F}_{A,(N_{k})}$. The measures $\mu$ and $\mu_{0}$
being nonnegative, replacing $f$ by its real part we can assume that $f$ is
real-valued, and thus that

$\int_{\mathbb{T}}f\,d\mu\leq\gamma_{1}<\gamma_{2}\leq\int_{\mathbb{T}^{d}}f\,d\mu_{0}\quad\textrm{for
every}\ \mu\in\boldsymbol{F}_{A,(N_{k})}.$ (4)

Moreover, it is possible to assume that $f$ is a Lipschitz map on
$\mathbb{T}^{d}=\mathbb{R}^{d}/\mathbb{Z}^{d}$, endowed with the distance
induced by the sup norm $||\,.\,||_{\infty,\mathbb{R}^{d}}$ on
$\mathbb{R}^{d}$. We thus suppose that there exists a constant $C>0$ such that

$|f(\boldsymbol{x}_{1})-f(\boldsymbol{x}_{2})|\leq
C\,\inf\left\\{||\boldsymbol{x}_{1}-\boldsymbol{x}_{2}-\boldsymbol{l}||_{\infty,\mathbb{R}^{d}}\;;\;\boldsymbol{l}\in\mathbb{Z}^{d}\right\\}\quad\textrm{for
every }\boldsymbol{x}_{1},\,\boldsymbol{x}_{2}\in\mathbb{T}^{d}.$ (5)

For any integer $k\geq 1$ and any element $\boldsymbol{x}_{k}$ of
$\mathbb{T}^{d}$ such that $(A^{{N_{k}}}-I)\boldsymbol{x}_{k}=0$, the measure
$\mu_{\boldsymbol{x}_{k}}=\frac{1}{N_{k}}\sum_{j=0}^{N_{k}-1}\delta_{\\{A^{{j}}\boldsymbol{x}_{k}\\}}$
belongs to $\boldsymbol{F}_{A,(N_{k})}$. Applying (4) to this measure yields
that

$\dfrac{1}{N_{k}}\sum_{j=0}^{N_{k}-1}f(A^{{j}}\boldsymbol{x}_{k})\leq\gamma_{1}<\gamma_{2}\leq\int_{\mathbb{T}^{d}}f\,d\mu_{0}.$
(6)

Let now $\boldsymbol{x}$ be an arbitrary element of $\mathbb{T}^{d}$, and let
$k\geq 1$. Consider the vector
$\boldsymbol{y}_{k}=(A^{N_{k}}-I)\,\boldsymbol{x}$, seen as an element of
$\mathbb{R}^{d}$ (and not as an element of $\mathbb{T}^{d}$). There exists
$\boldsymbol{l}_{k}\in\mathbb{Z}^{d}$ such that
$||\boldsymbol{y}_{k}-\boldsymbol{l}_{k}||_{\infty,\mathbb{R}^{d}}\leq 1$.
Recalling that $A=PDP^{-1}$, with
$D=\textrm{diag}(\lambda_{1},\dots,\lambda_{d})$, we thus have

$\displaystyle\bigl{|}\bigl{|}\,P(D^{N_{k}}-I)P^{-1}\boldsymbol{x}-\boldsymbol{l}_{k}\,\bigr{|}\bigr{|}_{\infty,\mathbb{R}^{d}}$
$\displaystyle\leq 1,$ so that
$\displaystyle\bigl{|}\bigl{|}\,(D^{N_{k}}-I)P^{-1}\boldsymbol{x}-P^{-1}\boldsymbol{l}_{k}\,\bigr{|}\bigr{|}_{\infty,\mathbb{C}^{d}}$
$\displaystyle\leq||P^{-1}||_{\infty}$ (7)

where $||P^{-1}||_{\infty}$ is the norm of the matrix $P^{-1}$ seen as an
endomorphism of $(\mathbb{C}^{d},||\,.\,||_{\infty,\mathbb{C}^{d}})$. The
inequality (7) means exactly that

$\sup_{1\leq i\leq
d}\bigl{|}\,(\lambda_{i}^{N_{k}}-1)\,(P^{-1}\boldsymbol{x})_{i}-(P^{-1}\boldsymbol{l}_{k})_{i}\bigr{|}\leq||P^{-1}||_{\infty}.$
(8)

Since no eigenvalue of $A$ belongs to the unit circle, $A^{N_{k}}-I$ is
invertible in $M_{d}(\mathbb{R})$ and it is legitimate to set
$\boldsymbol{z}_{k}=(A^{N_{k}}-I)^{-1}\boldsymbol{l}_{k}\in\mathbb{R}^{d}$.
Let $\boldsymbol{x}_{k}$ be the corresponding element of $\mathbb{T}^{d}$,
obtained by taking mod $1$ all the coordinates of $\boldsymbol{z}_{k}$. Then
$\boldsymbol{x}_{k}$ belongs to $\boldsymbol{C}_{A,(N_{k})}$, with
$(A^{N_{k}}-I)\boldsymbol{x}_{k}=0$ in $\mathbb{T}^{d}$. Also,
$\boldsymbol{z}_{k}=P(D^{N_{k}}-I)^{-1}P^{-1}\,\boldsymbol{l}_{k}$, so that

$P^{-1}\boldsymbol{z}_{k}=\Bigl{(}\dfrac{1}{\lambda_{i}^{N_{k}}-1}\cdot\bigl{(}P^{-1}\boldsymbol{l}_{k}\bigr{)}_{i}\Bigr{)}_{1\leq
i\leq d}.$

It follows from (8) that for every $i\in\\{1\ldots,d\\}$,

$\bigl{|}(P^{-1}\boldsymbol{x})_{i}-(P^{-1}\boldsymbol{z}_{k})_{i}\bigr{|}\leq\dfrac{||P^{-1}||_{\infty}}{|\lambda_{i}^{N_{k}}-1|}\cdot$
(9)

By (5),

$\displaystyle\bigl{|}\,f(A^{j}\boldsymbol{x})-f(A^{j}\boldsymbol{x}_{k})\,\bigr{|}$
$\displaystyle\leq
C\,\inf\left\\{||A^{j}\boldsymbol{x}-A^{j}\boldsymbol{x}_{k}-\boldsymbol{l}||_{\infty,\mathbb{R}^{d}}\;;\;\boldsymbol{l}\in\mathbb{Z}^{d}\right\\}$
$\displaystyle\leq
C\,\bigl{|}\bigl{|}A^{j}\boldsymbol{x}-A^{j}\boldsymbol{z}_{k}\bigr{|}\bigr{|}_{\infty,\mathbb{R}^{d}}$
$\displaystyle\leq
C\,||P||_{\infty}\,\bigl{|}\bigl{|}\,D^{j}(P^{-1}\boldsymbol{x}-P^{-1}\boldsymbol{z}_{k})\,\bigr{|}\bigr{|}_{\infty,\mathbb{C}^{d}}$
$\displaystyle=C\,||P||_{\infty}\,\sup_{1\leq i\leq
d}\,|\lambda_{i}^{j}|\,.\,\bigl{|}(P^{-1}\boldsymbol{x}-P^{-1}\boldsymbol{z}_{k})_{i}\bigr{|}$
$\displaystyle\leq
C\,||P||_{\infty}\,\sum_{i=1}^{d}\,|\lambda_{i}^{j}|\,.\,\bigl{|}(P^{-1}\boldsymbol{x}-P^{-1}\boldsymbol{z}_{k})_{i}\bigr{|}.$

Plugging into (9) yields that

$\bigl{|}\,f(A^{j}\boldsymbol{x})-f(A^{j}\boldsymbol{x}_{k})\,\bigr{|}\leq
C\,||P||_{\infty}\,.\,||P^{-1}||_{\infty}\,\sum_{i=1}^{d}\dfrac{|\lambda_{i}|^{j}}{|\lambda_{i}^{N_{k}}-1|}\cdot$

Hence

$\displaystyle\Bigl{|}\dfrac{1}{N_{k}}\sum_{j=0}^{N_{k}-1}f(A^{j}\boldsymbol{x})-\dfrac{1}{N_{k}}\sum_{j=0}^{N_{k}-1}f(A^{j}\boldsymbol{x}_{k})\Bigr{|}$
$\displaystyle\leq
C\,||P||_{\infty}\,||P^{-1}||_{\infty}\,\dfrac{1}{N_{k}}\sum_{i=1}^{d}\dfrac{1}{|\lambda_{i}^{N_{k}}-1|}\sum_{j=0}^{N_{k}-1}|\lambda_{i}|^{j}$
$\displaystyle\leq
C\,||P||_{\infty}\,||P^{-1}||_{\infty}\,\dfrac{1}{N_{k}}\sum_{i=1}^{d}\dfrac{|\lambda_{i}|^{N_{k}}-1}{|\lambda_{i}^{N_{k}}-1|\,(|\lambda_{i}|-1)}\cdot$

Notice that $|\lambda_{i}|\neq 1$ for all $i=1\ldots d$. Observe that
$(|\lambda_{i}|^{N_{k}}-1)/|\lambda_{i}^{N_{k}}-1|\longrightarrow 1$ as
${k}\longrightarrow{+\infty}$ if $|\lambda_{i}|>1$, while
$({|\lambda_{i}|^{N_{k}}-1)/|\lambda_{i}^{N_{k}}-1|\longrightarrow-1}$ as
${k}\longrightarrow{+\infty}$ if $|\lambda_{i}|<1$. We obtain the existence of
a positive constant $C^{\prime}$ such that

$\sup_{k\geq
1}\,\,\sum_{i=1}^{d}\dfrac{|\lambda_{i}|^{N_{k}}-1}{|\lambda_{i}^{N_{k}}-1|\,(|\lambda_{i}|-1)}\leq
C^{\prime}.$

Thus there exists $C^{\prime\prime}>0$ such that

$\Bigl{|}\dfrac{1}{N_{k}}\sum_{j=0}^{N_{k}-1}f(A^{j}\boldsymbol{x})-\dfrac{1}{N_{k}}\sum_{j=0}^{N_{k}-1}f(A^{j}\boldsymbol{x}_{k})\Bigr{|}\leq\dfrac{C^{\prime\prime}}{N_{k}}\quad\textrm{for
every}\ k\geq 1.$ (10)

The right hand bound in (10) tends to $0$ as $k$ tends to infinity. Combining
this with the fact that inequalities (6) and (10) hold true for every $k\geq
1$, we obtain that

$\limsup_{k\to+\infty}\dfrac{1}{N_{k}}\sum_{j=0}^{N_{k}-1}f(A^{j}\boldsymbol{x})\leq\gamma_{1}<\gamma_{2}\leq\int_{\mathbb{T}^{d}}f\,d\mu_{0}\quad\textrm{for
every }\boldsymbol{x}\in\mathbb{T}^{d}.$ (11)

Let $\varepsilon>0$ be such that $\gamma_{1}<\gamma_{2}-\varepsilon$. Applying
the Ergodic Decomposition Theorem to the measure $\mu_{0}$ yields the
existence of an ergodic $T_{A}$-invariant measure $\nu_{0}$ on
$\mathbb{T}^{d}$ such that

$\int_{\mathbb{T}^{d}}f\,d\nu_{0}\geq\int_{\mathbb{T}^{d}}f\,d\mu_{0}-\varepsilon.$

It then follows from (11) that

$\limsup_{k\to+\infty}\dfrac{1}{N_{k}}\sum_{j=0}^{N_{k}-1}f(A^{j}\boldsymbol{x})\leq\gamma_{1}<\gamma_{2}-\varepsilon\leq\int_{\mathbb{T}^{d}}f\,d\nu_{0}$
(12)

for every $\boldsymbol{x}\in\mathbb{T}^{d}$.

But since the measure $\nu_{0}$ is ergodic, the Birkhoff Pointwise Ergodic
Theorem implies that

$\limsup_{k\to+\infty}\dfrac{1}{N_{k}}\sum_{j=0}^{N_{k}-1}f(A^{j}\boldsymbol{x})=\int_{\mathbb{T}^{d}}f\,d\nu_{0}$

for $\nu_{0}$-almost every $\boldsymbol{x}\in\mathbb{T}^{d}$, which
contradicts (12). So the initial assumption that the set
$\mathcal{P}_{A}(\mathbb{T}^{d})\setminus\boldsymbol{F}_{A,(N_{k})}$ is non-
empty cannot hold, and Theorem 4.1 is proved ∎

###  4.b. Proof of Theorem 1.8

The proof of Theorem 1.8 is similar in spirit to that of Theorem 1.5. Of
course, assumption (H) and Theorem 2.1 are not available anymore, and they
have to be replaced by the following analogue of Fact 3.2:

######  Lemma 4.2.

Let $A\in M_{d}(\mathbb{Z})$ with $\det A\neq 0$ and $\det(A-I)\neq 0$, and
let $p_{1},\dots,p_{s}\geq 2$ be prime numbers such that
$\emph{gcd\,}(p_{i},\det A)=1$ for every $i\in\\{1,\ldots,s\\}$. There exist
an infinite subset $I$ of $\mathbb{N}$ and an integer $\gamma\geq 1$ such that
for every $i\in\\{1,\ldots,s\\}$ and every $N\in I$,

$\det\,\bigl{(}A^{N}-I\bigr{)}\not\equiv 0\mod p_{i}^{\gamma}.$

###### Proof of Lemma 4.2.

Since $\det(A-I)\neq 0$, there exists $\gamma\geq 1$ such that for every
$i\in\\{1,\ldots,s\\}$, $p_{i}^{\gamma}$ does not divide $\det(A-I)$. Since
$\textrm{gcd\,}(p_{i},\det A)=1$ and $p_{i}$ is prime,
$\textrm{gcd\,}(p_{i}^{\gamma},\det A)=1$ as well, and $A$ is invertible
modulo $p_{i}^{\gamma}$. Proceeding as in the proof of Theorem 1.7, we obtain
an integer $n_{i}\geq 2$ such that $A^{n_{i}}\equiv I\mod p_{i}^{\gamma}$, and
hence $A^{ln_{i}}\equiv I\mod p_{i}^{\gamma}$ for every $l\geq 1$. Setting
$n_{0}=n_{1}\dots n_{s}$, we have $A^{ln_{0}}\equiv I\mod p_{i}^{\gamma}$ for
every $l\geq 1$ and every $i\in\\{1,\ldots,s\\}$. Thus $A^{ln_{0}+1}-I\equiv
A-I\mod p_{i}^{\gamma}$ and $\det(A^{ln_{0}+1}-I)\equiv\det(A-I)\mod
p_{i}^{\gamma}$. Since $\det(A-I)\nequiv 0\mod p_{i}^{\gamma}$, we have
$\det(A^{ln_{0}+1}-I)\nequiv 0\mod p_{i}^{\gamma}$ for every $l\geq 1$ and
every $i\in\\{1,\ldots,s\\}$, and the lemma follows by setting
$I=n_{0}.\mathbb{N}+1$. ∎

Our aim is now to show that under the assumptions of Theorem 1.8, the
following fact holds:

######  Fact 4.3.

Suppose that $A,B\in M_{d}(\mathbb{Z})$ satisfy assumption (a), and either
assumption (b) or (b’) of Theorem 1.8. There exist a strictly increasing
sequence $(N_{k})_{k\geq 1}$ of integers and a finite subset $F$ of
$\mathbb{Z}\setminus\\{0\\}$ such that, for every $k\geq 1$, the integers
$q_{k}:=\det(A^{N_{k}}-I)$ can be decomposed as $q_{k}=h_{k}.r_{k}$, where
$h_{k}\in F$, $r_{k}\geq 1$, and $\textrm{gcd\,}(r_{k},\det B)=1$.

###### Proof of Fact 4.3.

Recall that since $A$ has no eigenvalue of modulus $1$, $A^{N_{k}}-I$ is
invertible in $M_{d}(\mathbb{C})$, and hence $q_{k}:=\det(A^{N_{k}}-I)\neq 0$.
If $\det B=\pm 1$, it suffices to choose $N_{k}=k$, $k\geq 1$, and $F=\\{\pm
1\\}$. So we suppose without loss of generality that $|\det B|\geq 2$. We
decompose $\det B$ as $\det B=\varepsilon\,p_{1}^{b_{1}}\dots p_{s}^{b_{s}}$,
where $\varepsilon=\pm 1$, $b_{i}\geq 1$ and $p_{i}$ is a prime number for
every $i\in\\{1,\ldots,s\\}$. We now treat separately two cases:

_Case_ 1: assumption (b) is satisfied, i.e. $\textrm{gcd\,}(\det A,\det B)=1$.
In this case $\textrm{gcd\,}(p_{i},\det A)=1$ for every
$i\in\\{1,\ldots,s\\}$, and Lemma 4.2 applies: there exist $\gamma\geq 1$ and
an infinite set $I\subseteq\mathbb{N}$ such that for every
$i\in\\{1,\ldots,s\\}$ and every $N\in I$, $p_{i}^{\gamma}$ does not divide
$\det(A^{N}-I)$. We enumerate the set $I$ as a strictly increasing sequence
$(N_{k})_{k\geq 1}$, and for each $k\geq 1$ we decompose
$q_{k}=\det(A^{N_{k}}-I)$ as $q_{k}=\varepsilon_{k}\,p_{1}^{a_{1,k}}\dots
p_{s}^{a_{s,k}}r_{k}$, where $\varepsilon_{k}=\pm 1$, $0\leq a_{i,k}<\gamma$
and $\textrm{gcd\,}(r_{k},p_{i})=1$ for each $i\in\\{1,\ldots,s\\}$. Setting

$F=\bigl{\\{}\pm p_{1}^{a_{1}}\dots p_{s}^{a_{s}}\;;\;0\leq
a_{i}<\gamma,\;\;i=1\ldots s\bigr{\\}}$

yields the conclusion of Fact 4.3 in this case.

_Case_ 2: assumption (b’) is satisfied. Let $a_{1},\dots,a_{d}$ be the
diagonal coefficients of $A$, which belong to $\mathbb{Z}\setminus\\{0\\}$.
For every $N\geq 1$, $\det(A^{N}-I)=\prod_{l=1}^{d}(a_{l}^{N}-1)$. By Fact 3.2
applied with $u=2$, there exist for each $l\in\\{1,\ldots,d\\}$ integers
$N_{1,l}\geq 3,\dots,N_{s,l}\geq 3$ as well as $\gamma_{l}\geq 1$ such that
for every $i\in\\{1,\ldots,s\\}$ and every
$N\in\mathbb{N}\setminus\bigcup_{i=1}^{s}N_{i,l}\,.\,\mathbb{N}$,

$|a_{l}|^{N}\not\equiv 1\mod p_{i}^{\gamma_{l}}.$

Since the integers $N_{i,l}$ are all greater or equal to $3$, the set
$J=\mathbb{N}\setminus\bigcup_{l=1}^{d}\bigcup_{i=1}^{s}N_{i,l}\,.\,\mathbb{N}$
contains an infinite subset $J^{\prime}$ consisting of _even_ integers. Let
$I=\\{M\geq 1\;;\;2M\in J^{\prime}\\}$. For every $i\in\\{1,\ldots,s\\}$,
every $l\in\\{1,\ldots,d\\}$ and every $M\in I$,

$a_{l}^{M}\not\equiv 1\mod p_{i}^{\gamma_{l}}.$

Setting $\gamma_{0}=\max_{1\leq l\leq d}\gamma_{l}$, we have thus

$a_{l}^{M}\not\equiv 1\mod p_{i}^{\gamma_{0}}$

for every $l\in\\{1,\ldots,d\\}$, $i\in\\{1,\ldots,s\\}$ and $M\in I$. We now
set $\gamma:=d\gamma_{0}$. Then $p_{i}^{\gamma}$ cannot divide the product
$\prod_{l=1}^{d}(a_{l}^{M}-1)$, since else $p_{i}^{\gamma_{0}}$ would divide
one of the terms $a_{l}^{M}-1$, $1\leq l\leq d$. Hence
$\det(A^{M}-I)\not\equiv 0\mod p_{i}^{\gamma}$ for every
$i\in\\{1,\ldots,s\\}$, and we conclude the proof as in the first case. ∎

We now use the notation from Fact 4.3. Since, for each $k\geq 1$, $r_{k}$ and
$\det B$ are relatively prime, $B$ is invertible modulo $r_{k}$, and there
exists an integer $m_{k}\geq 1$ such that $B^{m_{k}}\equiv I\mod r_{k}$ (see
the proof of Theorem 1.7). Hence $h_{k}\,B^{m_{k}}\equiv h_{k}\,I\mod q_{k}$
for every $k\geq 1$. If we define $h_{0}$ to be the product of all the
elements of the finite set $F$, it follows that

$h_{0}.B^{m_{k}}\equiv h_{0}.I\mod q_{k}\quad\textrm{for every}\ k\geq 1.$

Recall that given a measure $\mu\in\mathcal{P}(\mathbb{T}^{d})$ and a
$d$-tuple $\boldsymbol{n}=(n_{1},\dots,n_{d})\in\mathbb{Z}^{d}$, the
$\boldsymbol{n}$-th Fourier coefficient of the measure $\mu$ is defined as

$\widehat{\mu}(\boldsymbol{n})=\int_{\mathbb{T}^{d}}e^{2i\pi{\langle\boldsymbol{n},\boldsymbol{y}\rangle}}d\mu(\boldsymbol{y}),\quad\textrm{where}\
{\langle\boldsymbol{n},\boldsymbol{y}\rangle}=\sum_{i=1}^{d}n_{i}y_{i}.$

Write $\boldsymbol{h}_{0}=(h_{0},\dots,h_{0})$. Here is now an analogue of
Fact 2.7 in our multidimensional setting:

######  Fact 4.4.

Let $\boldsymbol{x}\in\mathbb{T}^{d}$ be such that
$(A^{N_{k}}-I)\boldsymbol{x}=0$ in $\mathbb{T}^{d}$ for some $k\geq 1$. Then

$\widehat{B^{lm_{k}}{\mu_{\boldsymbol{x}}}}(\boldsymbol{h}_{0})=\widehat{\mu}_{\boldsymbol{x}}(\boldsymbol{h}_{0})\quad\textrm{for
every integer}\ l\geq 1.$

###### Proof of Fact 4.4.

Recall that

$\mu_{\boldsymbol{x}}=\dfrac{1}{N_{k}}\sum_{j=0}^{N_{k}-1}\delta_{\\{A^{j}\boldsymbol{x}\\}}.$

For every $n\in\mathbb{Z}$,

$\displaystyle\widehat{B^{n}{\mu_{\boldsymbol{x}}}}(\boldsymbol{h}_{0})$
$\displaystyle=\dfrac{1}{N_{k}}\sum_{j=0}^{N_{k}-1}e^{2i\pi{\langle\boldsymbol{h}_{0},B^{n}A^{j}\boldsymbol{x}\rangle}}=\dfrac{1}{N_{k}}\sum_{j=0}^{N_{k}-1}e^{2i\pi{\langle\boldsymbol{1},h_{0}B^{n}A^{j}\boldsymbol{x}\rangle}}$
while $\displaystyle\widehat{\mu_{\boldsymbol{x}}}(\boldsymbol{h}_{0})$
$\displaystyle=\dfrac{1}{N_{k}}\sum_{j=0}^{N_{k}-1}e^{2i\pi{\langle\boldsymbol{1},h_{0}A^{j}\boldsymbol{x}\rangle}}$

where $\boldsymbol{1}=(1,\dots,1)$. Since $(A^{N_{k}}-I)\boldsymbol{x}=0$ in
$\mathbb{T}^{d}$, there exists $\boldsymbol{l}_{k}\in\mathbb{Z}^{d}$ such that
$(A^{N_{k}}-I)\boldsymbol{x}=\boldsymbol{l}_{k}$, the equality being this time
in $\mathbb{R}^{d}$, so that
$\boldsymbol{x}=\dfrac{1}{q_{k}}\,\textrm{adj}\,(A^{N_{k}}-I)\boldsymbol{l}_{k}$.

We know that $h_{0}\,B^{m_{k}}\equiv h_{0}\,I\ \textrm{mod}\ q_{k}$, so that
$h_{0}\,B^{lm_{k}}\equiv h_{0}\,I\ \textrm{mod}\ q_{k}$ for every $l\geq 1$.
This means that there exists a matrix $C_{k,l}\in M_{d}(\mathbb{Z})$ such that
$h_{0}.B^{lm_{k}}=h_{0}I+q_{k}.C_{l,k}$. Hence for every
$j\in\\{0,\ldots,N_{k}\\}$, we have

$h_{0}\,B^{lm_{k}}A^{j}\boldsymbol{x}=h_{0}\,A^{j}\boldsymbol{x}+q_{k}\,C_{k,l}A^{j}\boldsymbol{x}.$

Since
$\boldsymbol{x}=\dfrac{1}{q_{k}}\,\textrm{adj}\,(A^{N_{k}}-I)\boldsymbol{l}_{k}$
with $\boldsymbol{l}_{k}\in\mathbb{Z}^{d}$, the vector
$q_{k}C_{k,l}A^{j}\boldsymbol{x}$ belongs to $\mathbb{Z}^{d}$, and thus
$h_{0}\,B^{lm_{k}}A^{j}\boldsymbol{x}=h_{0}\,A^{j}\boldsymbol{x}$ in
$\mathbb{T}^{d}$. It follows that

$\widehat{B^{lm_{k}}\mu_{\boldsymbol{x}}}(\boldsymbol{h}_{0})=\widehat{\mu}_{\boldsymbol{x}}(\boldsymbol{h}_{0})\quad\textrm{for
every}\ l\geq 1,$

and Fact 4.4 is proved. ∎

A direct consequence of Fact 4.4 is that
$B^{n}\mu_{\boldsymbol{x}}\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
3.0pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 3.0pt\raise
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6.40012pt\raise 4.97223pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.97223pt\hbox{$\scriptstyle{w^{*}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 23.0pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces{\hbox{\kern
11.05365pt\raise-0.1pt\hbox{\hbox{\kern
0.0pt\raise-1.75pt\hbox{$\textstyle{\scriptstyle/}$}}}}}{\hbox{\kern
23.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}\ignorespaces}}}}\ignorespaces\textrm{Leb}_{d}$
when
$\textstyle{n\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{+\infty}$
as soon as $\widehat{\mu}_{\boldsymbol{x}}(\boldsymbol{h}_{0})\neq 0$.
Consider, for each $0<\gamma<1$, the set

$G_{A,B}^{\,\gamma}=\bigl{\\{}\mu\in\mathcal{P}_{A}(\mathbb{T}^{d})\;;\;\widehat{\mu}(\boldsymbol{h}_{0})\neq
0\ \textrm{and}\ \forall\,n_{0}\geq 1,\ \exists\,n\geq n_{0},\
|\widehat{B^{n}\mu}(\boldsymbol{h}_{0})|>\gamma|\widehat{\mu}(\boldsymbol{h}_{0})|\bigr{\\}}$

which is clearly a $G_{\delta}$ subset of
$(\mathcal{P}_{A}(\mathbb{T}^{d}),w^{*})$. In order to prove that it is dense,
we proceed as in the proof of Lemma 2.6, but using Theorem 4.1 instead of
Theorem 2.1. Let $\mathcal{V}$ be a non-empty open set in
$(\mathcal{P}_{A}(\mathbb{T}^{d}),w^{*})$. By Theorem 4.1, there exists a
convex combination

$\mu=\sum_{i=1}^{r}a_{i}\mu_{\boldsymbol{x}_{i}},\quad a_{i}\geq
0,\quad\sum_{i=1}^{r}a_{i}=1$

of measures $\mu_{\boldsymbol{x}_{i}}\in\boldsymbol{D}_{A,(N_{k})}$ which
belongs to $\mathcal{V}$.

Let $k_{i}$ be such that $(A^{N_{k_{i}}}-I)\boldsymbol{x}_{i}=0$, $1\leq i\leq
r$. By Fact 4.4,
$\widehat{B^{lm_{k_{i}}}\mu_{\boldsymbol{x}_{i}}}(\boldsymbol{h}_{0})=\widehat{\mu}_{\boldsymbol{x}_{i}}(\boldsymbol{h}_{0})$
for every $l\geq 1$. Setting $m_{0}=m_{k_{1}}\dots m_{k_{r}}$, we have that
$\widehat{B^{lm_{0}}\mu_{\boldsymbol{x}_{i}}}(\boldsymbol{h}_{0})=\widehat{\mu}_{\boldsymbol{x}_{i}}(\boldsymbol{h}_{0})$
for every $l\geq 1$ and every $i\in\\{1,\ldots,r\\}$. Hence
$\widehat{B^{lm_{0}}}(\boldsymbol{h}_{0})=\widehat{\mu}(\boldsymbol{h}_{0})$
for every $l\geq 1$.

If $\widehat{\mu}(\boldsymbol{h}_{0})\neq 0$, it follows that $\mu$ belongs to
$G_{A,B}^{\,\gamma}$. If $\widehat{\mu}(\boldsymbol{h}_{0})=0$, the measure
$\mu_{\rho}:=(1-\rho)\mu+\rho\delta_{\boldsymbol{0}}$ belongs to $\mathcal{V}$
if $0<\rho<1$ is sufficiently small, and
$\widehat{\mu}_{\rho}(\boldsymbol{h}_{0})=\rho\neq 0$. Also
$\widehat{B^{lm_{0}}\mu_{\rho}}(\boldsymbol{h}_{0})=(1-\rho)\widehat{B^{lm_{0}}\mu}(\boldsymbol{h}_{0})+\rho=\widehat{\mu_{\rho}}(\boldsymbol{h}_{0})$
for every $l\geq 0$, and hence $\mu_{\rho}$ belongs to $G_{A,B}^{\,\gamma}$.
We have thus shown that $G_{A,B}^{\,\gamma}$ is a dense $G_{\delta}$ subset of
$(\mathcal{P}_{A}(\mathbb{T}^{d}),w^{*})$.

Any measure $\mu\in G_{A,B}^{\,\gamma}$ is such that
$\limsup_{n\rightarrow+\infty}|\widehat{B^{n}\mu}(\boldsymbol{h}_{0})|>0$, and
hence (since $\boldsymbol{h}_{0}\neq\boldsymbol{0}$) such that
$B^{n}\mu\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
3.0pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 3.0pt\raise
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3.0pt}}}}}}\ignorespaces{\hbox{\kern 23.0pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces{\hbox{\kern
11.05365pt\raise-0.1pt\hbox{\hbox{\kern
0.0pt\raise-1.75pt\hbox{$\textstyle{\scriptstyle/}$}}}}}{\hbox{\kern
23.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}\ignorespaces}}}}\ignorespaces\textrm{Leb}_{d}$
for every $\mu\in G_{A,B}^{\,\gamma}$. So the set

$G_{A,B}^{\,0}=\bigl{\\{}\mu\in\mathcal{P}_{A}(\mathbb{T}^{d})\;;\;B^{n}\mu\lx@xy@svg{\hbox{\raise
0.0pt\hbox{\kern
3.0pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
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3.0pt}}}}}}\ignorespaces{\hbox{\kern 23.0pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces{\hbox{\kern
11.05365pt\raise-0.1pt\hbox{\hbox{\kern
0.0pt\raise-1.75pt\hbox{$\textstyle{\scriptstyle/}$}}}}}{\hbox{\kern
23.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}\ignorespaces}}}}\ignorespaces\textrm{Leb}_{d}\bigr{\\}}$

is residual in $(\mathcal{P}_{A}(\mathbb{T}^{d}),w^{*})$.

In order to complete the proof, it remains to show the following analogue of
Fact 2.8:

######  Fact 4.5.

The set $\mathcal{P}_{A,c}(\mathbb{T}^{d})$ is a dense $G_{\delta}$ subset of
$(\mathcal{P}_{A}(\mathbb{T}^{d}),w^{*})$.

###### Proof of Fact 4.5.

As mentioned already in the proof of Fact 2.8, the set
$\mathcal{P}_{A,c}(\mathbb{T}^{d})$ is known to be a $G_{\delta}$ subset of
$(\mathcal{P}_{A}(\mathbb{T}^{d}),w^{*})$, so that only its density remains to
be proved. The argument for this follows closely the proof of [S1, Th. 2], and
reproves at the same time that $\mathcal{P}_{A,c}(\mathbb{T}^{d})$ is
$G_{\delta}$.

For every $\tau>0$, let

$F_{\tau}:=\bigl{\\{}\mu\in\mathcal{P}_{A}(\mathbb{T}^{d})\;;\;\exists\,\boldsymbol{x}\in\mathbb{T}^{d}\
\textrm{such that}\ \mu(\\{\boldsymbol{x}\\})\geq\tau\bigr{\\}}.$

Then the set $F_{\tau}$ is easily seen to be closed in
$(\mathcal{P}_{A}(\mathbb{T}^{d}),w^{*})$. Let us now show that $F_{\tau}$ is
nowhere dense. Let $(N_{k})_{k\geq 1}$ be a strictly increasing sequence of
_prime numbers_ such that $N_{1}>1/\tau$. By Theorem 4.1, the convex hull of
the set

$\boldsymbol{D}_{A,(N_{k})}=\bigl{\\{}\mu_{\boldsymbol{x}}\;;\;A^{N_{k}}\boldsymbol{x}=\boldsymbol{x}\
\textrm{for some}\ k\geq 1\bigr{\\}}$

is dense in $(\mathcal{P}_{A}(\mathbb{T}^{d}),w^{*})$. Hence, given a non-
empty open subset $\mathcal{V}$ of $(\mathcal{P}_{A}(\mathbb{T}^{d}),w^{*})$,
there exist vectors $\boldsymbol{x}_{1},\dots,\boldsymbol{x}_{r}$ in
$\mathbb{T}^{d}$, integers $k_{1},\dots,k_{r}\geq 1$ and coefficients
$a_{1},\dots,a_{r}\geq 0$ with $\sum_{i=1}^{r}a_{i}=1$ and
$A^{N_{k_{i}}}\boldsymbol{x}_{i}=\boldsymbol{x}_{i}$ for each $1\leq i\leq r$
such that

$\mu=\sum_{i=1}^{r}a_{i}\mu_{\boldsymbol{x}_{i}}\quad\textrm{belongs to}\
\mathcal{V}.$

Since $N_{k_{i}}$ is prime, the minimal period of $\boldsymbol{x}_{i}$ is
$N_{k_{i}}$, and thus

$\mu_{\boldsymbol{x}_{i}}(\\{\boldsymbol{x}\\})\leq\dfrac{1}{N_{k_{i}}}<{\tau}\quad\textrm{for
every}\ \boldsymbol{x}\in\mathbb{T}^{d}.$

It follows that $\mu(\\{\boldsymbol{x}\\})<\tau$ for every
$\boldsymbol{x}\in\mathbb{T}^{d}$, and $\mu$ does not belong to $F_{\tau}$. So
$F_{\tau}$ is nowhere dense in $(\mathcal{P}_{A}(\mathbb{T}^{d}),w^{*})$, and

$\mathcal{P}_{A,c}(\mathbb{T}^{d})=\mathcal{P}_{A}(\mathbb{T}^{d})\setminus\bigcup_{l\geq
1}F_{2^{-l}}$

is a dense $G_{\delta}$ subset of $(\mathcal{P}_{A}(\mathbb{T}^{d}),w^{*})$ by
the Baire Category Theorem. ∎

The proof of Theorem 1.8 is completed by combining Fact 4.5 with the assertion
that $G_{A,B}^{0}$ is residual in $(\mathcal{P}_{A}(\mathbb{T}^{d}),w^{*})$.
$\square$

######  Remark 4.6.

The proof of Fact 4.5 would apply equally well to Fact 2.8, but since the
result is more standard in the one-dimensional case, we preferred to mention
the classical arguments.

##  5\. Further results and remarks

###  5.a. A complement to a result of Johnson and Rudolph

Let $p\geq 2$ be an integer, and let $(c_{n})_{n\geq 0}$ be a sequence of
positive integers. We have recalled in the introduction and in Section 3
conditions on $(c_{n})$ implying that each measure
$\mu\in\mathcal{P}_{p}(\mathbb{T})$ which is ergodic and of positive entropy
is $(c_{n})$-generic — thus showing that the set

$G^{\prime}_{p,(c_{n})}:=\bigl{\\{}\mu\in\mathcal{P}_{p}(\mathbb{T})\;;\;T_{c_{n}}\mu\xrightarrow{\text{$w^{*}$}}\textrm{Leb}\quad\textrm{along
a sequence of upper density}\ 1\bigr{\\}}$

is residual in $(\mathcal{P}_{p}(\mathbb{T}),w^{*})$. We present here an
alternative harmonic analysis approach to this kind of result. It has the
benefit of circumventing the arguments that depend on positive entropy, when
applicable.

######  Theorem 5.1.

Let $p\geq 2$, and let ${(c_{n})}_{n\geq 0}$ be a sequence of integers
satisfying the following condition:

$(\star)$ there exists a sequence $(\mu_{k})_{k\geq 1}$ of elements of
$\mathcal{P}_{p}(\mathbb{T})$ such that
$\mu_{k}\xrightarrow{\text{$w^{*}$}}\delta_{1}$, and moreover, the set
$\bigl{\\{}n\geq 1\;;\;|\widehat{\mu}_{k}(a.c_{n})|<\varepsilon\bigr{\\}}$ has
density $1$ for every $a\in\mathbb{Z}\setminus\\{0\\}$, every $\varepsilon>0$
and every $k\geq 1$.

Then the set $G^{\prime}_{p,(c_{n})}$ is residual in
$(\mathcal{P}_{p}(\mathbb{T}),w^{*})$.

A word about terminology: saying that a sequence $(\nu_{n})_{n\geq 1}$ of
measures converges to $\nu$ along a subset of upper density $1$ means that for
any neighborhood $\mathcal{V}$ of $\nu$ in $\mathcal{P}(\mathbb{T})$, the set
$\\{n\geq 1\;;\;\nu_{n}\in\mathcal{V}\\}$ has upper density $1$, i. e.

$\limsup_{N\to+\infty}\,\dfrac{1}{N}\,\\#\,\bigl{\\{}1\leq n\leq
N\;;\;\nu_{n}\in\mathcal{V}\bigr{\\}}=1.$

This is equivalent to the following property: for any $a_{0}\geq 1$ and any
$\varepsilon>0$, the set

$\bigl{\\{}n\geq
1\;;\;|\widehat{\nu}_{n}(a)-\widehat{\nu}(a)|<\varepsilon\quad\textrm{for
every}\ a\in\mathbb{Z}\ \textrm{with}\ |a|\leq a_{0}\bigr{\\}}$

has upper density $1$. In this case, one can construct a strictly increasing
sequence $(N_{k})_{k\geq 1}$ of integers such that

$\dfrac{1}{N_{k}}\,\\#\,\bigl{\\{}1\leq n\leq N_{k}\;;\;\forall\,|a|\leq k,\
|\widehat{\nu}_{n}(a)-\widehat{\nu}(a)|<2^{-k}\bigr{\\}}\geq 1-2^{-k}$

for every $k\geq 1$, and $N_{k+1}\geq 2^{k}N_{k}$. If we consider the strictly
increasing sequence $(n_{j})_{j\geq 1}$ obtained by enumerating the set

$D=\bigcup_{k\geq 1}\bigl{\\{}N_{k-1}<n\leq N_{k}\;;\;\forall\,|a|\leq k,\
|\widehat{\nu}_{n}(a)-\widehat{\nu}(a)|<2^{-k}\bigr{\\}}$

(with the convention that $N_{0}=0$), we obtain that $D=\\{n_{j}\;;\;j\geq
1\\}$ has upper density $1$ and that
$\widehat{\nu}_{n_{j}}(a)\longrightarrow\widehat{\nu}(a)$ as
$j\longrightarrow+\infty$ for every $a\in\mathbb{Z}$.

###### Proof of Theorem 5.1.

We first observe that the set $G^{\prime}_{p,(c_{n})}$ can be written as
$G^{\prime}_{p,(c_{n})}=\widetilde{G}_{p,(c_{n})}\cap\mathcal{P}_{p,c}(\mathbb{T})$,
where

$\displaystyle\widetilde{G}_{p,(c_{n})}=\bigl{\\{}$
$\displaystyle\mu\in\mathcal{P}_{p}(\mathbb{T})\;;\;\forall\,N_{0},a_{0}\geq
1,\quad\forall\,\varepsilon,\delta\in(0,1)\cap\mathbb{Q}$
$\displaystyle\exists\,N>N_{0},\quad\exists\,F\subseteq\\{1,\dots,N\\}\quad\textrm{with}\
\\#\,F\geq(1-\delta)N$ $\displaystyle\textrm{such that}\
\forall\,a\in\mathbb{Z}\ \textrm{with}\ 0<|a|\leq a_{0},\ \forall\,n\in
F,\;|\widehat{\mu}(a.c_{n})|<\varepsilon\bigr{\\}}.$

The set $\widetilde{G}_{p,(c_{n})}$ is clearly $G_{\delta}$ in
$(\mathcal{P}_{p}(\mathbb{T}),w^{*})$. Since $\mathcal{P}_{p,c}(\mathbb{T})$
is residual in $\mathcal{P}_{p}(\mathbb{T})$, it suffices to show that
$\widetilde{G}_{p,(c_{n})}$ is dense in $\mathcal{P}_{p}(\mathbb{T})$. In
order to do this, we are going to exhibit a dense set of measures
$\mu\in\mathcal{P}_{p}(\mathbb{T})$ with the following property:

$\forall\,a\in\mathbb{Z}\setminus\\{0\\},\ \forall\,\varepsilon>0,\
\textrm{the set}\ \\{n\geq 1\;;\;|\widehat{\mu}(ac_{n})|<\varepsilon\\}\
\textrm{has density 1.}$ (13)

Since the intersection of finitely many sets of density $1$ is again of
density $1$, the measures in this set will be such that

$\forall\,a_{0}\geq 1,\ \forall\,\varepsilon>0,\ \textrm{the set}\ \\{n\geq
1\;;\;\forall\,0<|a|\leq a_{0},\ |\widehat{\mu}(ac_{n})|<\varepsilon\\}\
\textrm{has density 1}$ (14)

and hence upper density $1$. Such measures will hence belong to the set
$G^{\prime}_{p,(c_{n})}$.

Our assumption $(\star)$ states that the measures $\mu_{k}$, $k\geq 1$,
satisfy (13). Fix $\nu\in\mathcal{P}_{p}(\mathbb{T})$, and set
$\nu_{k}=\mu_{k}*\nu$ for every $n\geq 1$. For any $\varepsilon>0$, the set
$\\{n\geq 1\;;\;|\widehat{\nu}_{k}(a.c_{n})|<\varepsilon\\}$ has density $1$,
and it follows that the measures $\nu_{k}$ satisfy (13). Since
$\nu_{k}\xrightarrow{\text{$w^{*}$}}\nu$ as
$\textstyle{k\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{+\infty,}$
this concludes the proof of Theorem 5.1. ∎

Theorem 5.1 applies for instance to the case where $c_{n}=q^{n}$, $n\geq 0$,
provided that $p,q\geq 2$ are two multiplicatively independent integers, and
allows to retrieve [JR, Th. 8.2], which states that $G^{\prime}_{p,(q^{n})}$
is residual in $(\mathcal{P}_{p}(\mathbb{T}),w^{*})$.

To this aim, it suffices to exhibit a sequence $(\mu_{k})_{k\geq 1}$ of
measures from $\mathcal{P}_{p}(\mathbb{T})$ satisfying (13) and such that
$\mu_{k}\xrightarrow{\text{$w^{*}$}}\delta_{1}$. The measures that we shall
consider are the Bernoulli convolutions $\mu_{\Theta}$ introduced at the end
of the proof of Theorem 1.5, where $\Theta=(\theta_{0},\dots,\theta_{p-1})$ is
a $p$-tuple of elements of $(0,1)$ with $\sum_{j=0}^{p-1}\theta_{j}=1$. They
are $T_{p}$-invariant, and

$\widehat{\mu}_{\Theta}(m)=\prod_{n\geq
1}\Bigl{(}\theta_{0}+\sum_{j=1}^{p-1}\theta_{j}e^{2i\pi
mjp^{-n}}\Bigr{)}\quad\textrm{for every }m\in\mathbb{Z}.$

It is shown by Lyons in [L1] and by Feldman and Smorodinsky in [FS] that
$T_{q^{n}}\mu_{\Theta}\xrightarrow{\text{$w^{*}$}}\textrm{Leb}$ as
$\textstyle{n\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{+\infty.}$
Since $\mu_{\Theta}\xrightarrow{\text{$w^{*}$}}\delta_{1}$ as
$\textstyle{\Theta\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(1,0,\dots,0),}$
assumption $(\star)$ from Theorem 5.1 is satisfied, and
$G^{\prime}_{p,(q^{n})}$ is residual in $(\mathcal{P}_{p}(\mathbb{T}),w^{*})$.

The behaviour of the Fourier coefficients of Bernoulli convolutions has been
studied extensively, particularly when $p$ is equal to $2$ or $3$; see for
instance the classical book [KS] by Kahane and Salem. An important work on the
subject is that of Blum and Epstein [BE], where the authors provide upper and
lower bounds on $|\widehat{\mu}_{\Theta}(m)|^{2}$ which allow them to give a
characterisation of sequences of positive integers along which
$\widehat{\mu}_{\Theta}(m)$ tends to $0$. In the case $p=2$, this
characterisation is given in terms of the order of magnitude of $R(m)$, which
is the number of _runs_ , _i. e._ of maximal blocks of the same digit $0$ or
$1$, appearing in the binary expansion of $m$. Equivalently, $R(m)$ is the
number of digits changes in the binary expansion of $m$. Then as soon as
$\Theta\neq(1/2,1/2)$ (in which case $\mu_{\Theta}$ is the Lebesgue measure on
$\mathbb{T}$), there exist two constants $C_{1},C_{2}>0$ such that, for every
$m\in\mathbb{Z}$,

$\exp(-C_{2}R(m))\leq|\widehat{\mu}_{\Theta}(m)|\leq\exp(-C_{1}R(m)).$

It follows that if $(m_{k})$ is any strictly increasing sequence of integers,
we have $\widehat{\mu}_{\Theta}(m_{k})\longrightarrow 0$ as
$k\longrightarrow+\infty$ if and only if $R(m_{k})\longrightarrow+\infty$ as
$k\longrightarrow+\infty$.

For general $p$, here is the result proved by Blum and Epstein in [BE]:

######  Theorem 5.2 ([BE]).

Let $p\geq 2$. Let $\Theta=(\theta_{0},\dots,\theta_{p-1})$ be a $p$-tuple of
elements from $(0,1)$ with the property that the polynomial
$Q_{\Theta}(z)=\sum_{j=0}^{p-1}\theta_{j}z^{j}$ does not vanish on
$\mathbb{T}$. Let, for $m\geq 0$,

$\psi(m)=R_{0}(m)+R_{p-1}(m)+N(m),$

where $R_{0}(m)$ is the number of maximal blocks of $0$s, $R_{p-1}(m)$ is the
number of maximal blocks of $(p-1)$s, and $N(m)$ is the number of digits other
than $0$ and $p-1$ in the expansion of $m$ in base $p$. Then there exist two
constants $C_{1},C_{2}>0$ such that

$\exp(-C_{2}\psi(m))\leq|\widehat{\mu}_{\Theta}(m)|\leq\exp(-C_{1}\psi(m))\quad\textrm{for
every}\ m\in\mathbb{Z}.$

Hence $\widehat{\mu}_{\Theta}(m_{k})\longrightarrow 0$ as
$k\longrightarrow+\infty$ if and only if $\psi(m_{k})\longrightarrow+\infty$
as $k\longrightarrow+\infty$.

As a consequence, we obtain the following result, which we state using the
notation from Theorem 5.2:

######  Proposition 5.3.

Let $(c_{n})_{n\geq 0}$ be a sequence of integers such that, for every
$a\in\mathbb{Z}\setminus\\{0\\}$, the sequence $(\psi(a.c_{n}))_{n\geq 0}$
tends to infinity along a sequence of density $1$. Then the set
$G^{\prime}_{p,(c_{n})}$ is residual in $(\mathcal{P}_{p}(\mathbb{T}),w^{*})$.

###### Proof.

It suffices to show that $(c_{n})_{n\geq 0}$ satisfies assumption $(\star)$ of
Theorem 5.1. Let $\Theta=(\theta_{0},\dots,\theta_{p-1})$ be a sequence of
elements of $(0,1)$ summing up to $1$, and suppose that $\theta_{0}>1/2$. Then
$\sum_{j=1}^{p-1}\theta_{j}<1/2$, and hence we have

$|Q_{\Theta}(z)|\geq\theta_{0}-\sum_{j=1}^{p-1}\theta_{j}|z|^{j}>0$

for every $z\in\mathbb{C}$ with $|z|\leq 1$. Thus the polynomial $Q_{\Theta}$
does not vanish on $\mathbb{T}$. Let $(\Theta_{k})_{k\geq 1}$ be such that
$\textstyle{\Theta_{k}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(1,0,\dots,0)}$
as
$\textstyle{k\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{+\infty,}$
and $Q_{\Theta_{k}}$ does not vanish on $\mathbb{T}$. Then
$\mu_{\Theta_{k}}\xrightarrow{\text{$w^{*}$}}\delta_{1}$ as
$\textstyle{k\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{+\infty}$.
Moreover, the assumption of Proposition 5.3 combined with Theorem 5.2 implies
that for every $a\in\mathbb{Z}\setminus\\{0\\}$,
$\textstyle{\widehat{\mu}_{\Theta_{k}}(ac_{n})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$
along a sequence of density $1$. The assumption $(\star)$ is thus satisfied,
and Proposition 5.3 follows. ∎

######  Remark 5.4.

If $p,q\geq 2$ are multiplicatively independent integers, and $c_{n}=q^{n}$,
$n\geq 0$, it is shown in [FS, Prop. 1] that any word $w$ in the letters
$0,1,\dots,p-1$ appears in the expansion of $q^{n}$ in base $p$ for every $n$
belonging to a set of integers of density $1$. The proof given there can be
extended to show that for any $a\in\mathbb{Z}\setminus\\{0\\}$, $w$ appears in
the expansion of $a.q^{n}$ in base $p$ for every $n$ belonging to a set of
integers of density $1$.

###  5.b. Some open questions

In Sections 2 and 3, numerous examples of sequences $(c_{n})$ were presented,
for which the set

$G_{p,(c_{n})}{}=\bigl{\\{}\mu\in\mathcal{P}_{p,c}(\mathbb{T})\;;\;T_{c_{n}}\mu\lx@xy@svg{\hbox{\raise
0.0pt\hbox{\kern
3.0pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
6.40012pt\raise 4.97223pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.97223pt\hbox{$\scriptstyle{w^{*}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 23.0pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces{\hbox{\kern
11.05365pt\raise-0.1pt\hbox{\hbox{\kern
0.0pt\raise-1.75pt\hbox{$\textstyle{\scriptstyle/}$}}}}}{\hbox{\kern
23.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}\ignorespaces}}}}\ignorespaces\textrm{Leb}\bigr{\\}}$

was found to be a residual subset of $(\mathcal{P}_{p}(\mathbb{T}),w^{*})$.
The condition (H) presented in Section 2 is the most general one that we can
provide for the residuality of $G_{p,(c_{n})}{}$ to hold. However, it does not
apply to _all_ sequences $(c_{n})_{n\geq 0}$, leaving the following intriguing
question unanswered.

######  Question 5.5.

Let $p\geq 2$, and let $(c_{n})_{n\geq 0}$ be any strictly increasing sequence
of integers. Is it true that the set $G_{p,(c_{n})}{}$ is residual in
$(\mathcal{P}_{p}(\mathbb{T}),w^{*})$?

If $\mu$ is $(c_{n})$-generic, then
$\frac{1}{N}\sum_{n=0}^{N-1}T_{c_{n}}\mu\xrightarrow{\text{$w^{*}$}}\textrm{Leb}\quad\textrm{as}\
\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
7.56248pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{N\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
27.56248pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
27.56248pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{+\infty}$}}}}}\ignorespaces}}}}\ignorespaces.$ It is
thus natural to consider the set

$G_{p,(c_{n})}^{\prime\prime}=\bigl{\\{}\mu\in\mathcal{P}_{p,c}(\mathbb{T})\;;\;\dfrac{1}{N}\sum_{n=0}^{N-1}T_{c_{n}}\mu\xrightarrow{\text{$w^{*}$}}\textrm{Leb}\quad\textrm{as
}N\longrightarrow+\infty\bigr{\\}}$

and to ask the following question.

######  Question 5.6.

Let $p\geq 2$, and let $(c_{n})_{n\geq 0}$ be a strictly increasing sequence
of integers. Is the set $G_{p,(c_{n})}^{\prime\prime}$ residual in
$(\mathcal{P}_{p}(\mathbb{T}),w^{*})$?

For the sequences considered in [H, JR, M, LMP], the density of the set
$G_{p,(c_{n})}^{\prime\prime}{}$ in $(\mathcal{P}_{p}(\mathbb{T}),w^{*})$
follows from the result that ergodic measures of positive entropy in
$\mathcal{P}_{p}(\mathbb{T})$ are $(c_{n})$-generic. But the question of the
residuality remains widely open, and it is actually not known if the set
$G_{p,(q^{n})}^{\prime\prime}{}$ is residual in
$(\mathcal{P}_{p}(\mathbb{T}),w^{*})$ when $q\geq 2$ is an integer which is
multiplicatively independent from $p$. This question is also connected to
another conjecture from [L2], called (C7), which runs as follows:

Conjecture (C7): _Let $p,q\geq 2$ be multiplicatively independent integers.
For any measure $\mu\in\mathcal{P}_{p,c}(\mathbb{T})$, $\mu$-almost every
$x\in\mathbb{T}$ is normal in base $q$, i.e._

$\dfrac{1}{N}\sum_{n=0}^{N-1}e^{2i\pi aq^{n}x}\lx@xy@svg{\hbox{\raise
0.0pt\hbox{\kern
3.0pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
23.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
23.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}\ignorespaces}}}}\ignorespaces{0}\quad\textrm{as}\
\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
7.56248pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{N\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
27.56248pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
27.56248pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{0}$}}}}}\ignorespaces}}}}\ignorespaces\ \textrm{for
every}\ a\in\mathbb{Z}\setminus\\{0\\}.$

In our examination of Conjecture 1.4 in the multidimensional context, we have
established conditions on matrices $A,B\in M_{d}(\mathbb{Z})$ with non-zero
determinant that imply that the set

$G_{A,B}{}=\bigl{\\{}\mu\in\mathcal{P}_{A,c}(\mathbb{T})\;;\;T_{B^{n}}\mu\lx@xy@svg{\hbox{\raise
0.0pt\hbox{\kern
3.0pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
6.40012pt\raise 4.97223pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.97223pt\hbox{$\scriptstyle{w^{*}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 23.0pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces{\hbox{\kern
11.05365pt\raise-0.1pt\hbox{\hbox{\kern
0.0pt\raise-1.75pt\hbox{$\textstyle{\scriptstyle/}$}}}}}{\hbox{\kern
23.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}\ignorespaces}}}}\ignorespaces\textrm{Leb}_{d}\bigr{\\}}$

is residual in $(\mathcal{P}_{A}(\mathbb{T}^{d}),w^{*})$. These conditions
(that $A$ be diagonalisable in $M_{d}(\mathbb{C})$, that $\det A$ and $\det B$
be relatively prime…) arise due to technical difficulties in the proofs in the
higher dimensional case. However, it may be that these conditions are not
necessary; this is true in the one-dimensional setting.

######  Question 5.7.

Let $d\geq 2$ and $A,B\in M_{d}(\mathbb{Z})$ with $\det A\neq 0$ and $\det
B\neq 0$. Is it true that the set $G_{A,B}$ is residual in
$(\mathcal{P}_{A}(\mathbb{T}^{d}),w^{*})$?

Let also

$G_{A,B}^{\prime\prime}=\bigl{\\{}\mu\in\mathcal{P}_{A,c}(\mathbb{T}^{d})\;;\;\dfrac{1}{N}\sum_{n=0}^{N-1}T_{B^{n}}\mu\xrightarrow{\text{$w^{*}$}}\textrm{Leb}_{d}\quad\textrm{as
}N\longrightarrow+\infty\bigr{\\}}.$

In analogy to Question 5.6 , one may also ask:

######  Question 5.8.

Let $d\geq 2$ and $A,B\in M_{d}(\mathbb{Z})$ with $\det A\neq 0$ and $\det
B\neq 0$. Is it true that the set $G_{A,B}^{\prime\prime}$ is residual in
$(\mathcal{P}_{A}(\mathbb{T}^{d}),w^{*})$?

## References