proof

# Averages of completely multiplicative functions over the Gaussian integers –
a dynamical approach

Sebastián Donoso  ,  Anh N. Le  ,  Joel Moreira  and  Wenbo Sun  Departamento
de Ingenier\́mathrm{i}a Matemática and Centro de Modelamiento Matemático,
Universidad de Chile & IRL-CNRS 2807, Beauchef 851, Santiago, Chile.
<EMAIL_ADDRESS>Department of Mathematics
University of Denver
2390 S. York St., Denver, CO, 80210, USA<EMAIL_ADDRESS>Mathematics
Institute
University of Warwick
Coventry, UK<EMAIL_ADDRESS>Department of Mathematics
Virginia Polytechnic Institute and State University
225 Stanger Street, Blacksburg, VA, 24061-1026, USA<EMAIL_ADDRESS>
###### Abstract.

We prove a pointwise convergence result for additive ergodic averages
associated with certain multiplicative actions of the Gaussian integers. We
derive several applications in dynamics and number theory, including:

  1. (i)

Wirsing’s theorem for Gaussian integers: if $f\colon\mathbb{G}\to\mathbb{R}$
is a bounded completely multiplicative function, then the following limit
exists:

$\lim_{N\to\infty}\frac{1}{N^{2}}\sum_{1\leq m,n\leq N}f(m+\mathrm{i}n).$

  2. (ii)

An answer to a special case of a question of Frantzikinakis and Host: for any
completely multiplicative real-valued function $f:\mathbb{N}\to\mathbb{R}$,
the following limit exists:

$\lim_{N\to\infty}\frac{1}{N^{2}}\sum_{1\leq m,n\leq N}f(m^{2}+n^{2}).$

  3. (iii)

A variant of a theorem of Bergelson and Richter on ergodic averages along the
$\Omega$ function: if $(X,T)$ is a uniquely ergodic system with unique
invariant measure $\mu$, then for any $x\in X$ and $f\in C(X)$,

$\lim_{N\to\infty}\frac{1}{N^{2}}\sum_{1\leq m,n\leq
N}f(T^{\Omega(m^{2}+n^{2})}x)=\int_{X}f\ d\mu.$

S.D. was partially funded by Centro de Modelamiento Matemático (CMM) FB210005
BASAL funds for centers of excellence from ANID-Chile and
ANID/Fondecyt/1200897. W.S. was partially supported by the NSF Grant
DMS-2247331

###### Contents

  1. 1 Introduction
    1. 1.1 Wirsing theorems for Gaussian integers
    2. 1.2 Ergodic averages along $\Omega(m^{2}+n^{2})$ in uniquely ergodic systems
    3. 1.3 The main theorem
    4. 1.4 Outline of the article
  2. 2 Background
    1. 2.1 Notation
    2. 2.2 Distribution of Gaussian primes
    3. 2.3 Dilated Følner sequences
  3. 3 Proof of E
    1. 3.1 Roadmap of the proof of Theorem 3.1
    2. 3.2 Proof of Lemma 3.2
    3. 3.3 Proof of Lemma 3.6
    4. 3.4 Sparse sets
    5. 3.5 Proof of Lemma 3.4
    6. 3.6 Proof of Lemma 3.5
  4. 4 Applications
    1. 4.1 Comparing the averages of two multiplicative functions
    2. 4.2 Proof of C
    3. 4.3 Proofs of Theorems A, 1.2 and B
    4. 4.4 Proof of D
  5. 5 Open questions
  6. A Some estimates
  7. B A counterexample with non-dilated Følner sequences

##  1\. Introduction

###  1.1. Wirsing theorems for Gaussian integers

A function $f:\mathbb{N}\to\mathbb{C}$ is called _multiplicative_ if
$f(mn)=f(m)f(n)$ whenever $m,n\in\mathbb{N}$ are co-prime; and $f$ is
_completely multiplicative_ if this relation holds for every
$m,n\in\mathbb{N}$. The statistical behavior of multiplicative functions is a
central topic in analytic number theory and many important theorems in this
area can be recast into the language of multiplicative functions. For
instance, the Liouville function $\lambda:\mathbb{N}\to\\{-1,1\\}$ defined as
$\lambda(n)=(-1)^{\Omega(n)}$, where $\Omega(n)$ is the number of prime
factors of $n$ counting with multiplicities, is a completely multiplicative
function. It is well known that the Prime Number theorem is equivalent to

$\lim_{N\to\infty}\frac{1}{N}\sum_{n=1}^{N}\lambda(n)=0$ (1.1)

(see, for example, [2, Page 96]). The harder part of establishing (1.1) is to
show that the limit exists. Generalizing this fact, Erdős and Wintner [8]
conjectured that for any multiplicative function $f:\mathbb{N}\to\\{-1,1\\}$,
the limit

$\lim_{N\to\infty}\frac{1}{N}\sum_{n=1}^{N}f(n)$ (1.2)

exists. Around 1961, theorems of Delange [6] and Wirsing [30] gave a
satisfactory answer for multiplicative functions with non-zero mean values.
The general case of the Erdős-Wintner conjecture, which contains a proof of
the Prime Number Theorem, was only established by Wirsing [31] in 1967. In
fact, Wirsing’s theorem states that the limit (1.2) exists for any bounded
real valued multiplicative function. A celebrated result of Halász [16] in
1968 further extended the analysis to complex valued functions, where the
picture is complicated by the fact that the limit (1.2) does not always exist.

There are several possible ways to strengthen (1.1) or Wirsing’s theorem. For
example, a conjecture by Chowla [2, Page 96] states that if
$P\in\mathbb{Z}[x]$ is a polynomial satisfying $P\neq cQ^{2}$ for every
$c\in\mathbb{Z},Q\in\mathbb{Z}[x]$, then

$\lim_{N\to\infty}\frac{1}{N}\sum_{n=1}^{N}\lambda(P(n))=0.$ (1.3)

This conjecture is still wide open despite a large number of developments seen
in the last decade. A survey of much of the progress can be found in [9] and
references therein.

Averages of multiplicative functions over one variable, such as in (1.3), are
notoriously hard to analyze. However, their multivariate counterpart seems to
be more tractable. In this direction, another conjecture, also attributed to
Chowla states that if $P\in\mathbb{Z}[x,y]$ and $P\neq cQ^{2}$ for every
$c\in\mathbb{Z},Q\in\mathbb{Z}[x,y]$ then

$\lim_{N\to\infty}\frac{1}{N^{2}}\sum_{m,n=1}^{N}\lambda(P(m,n))=0.$ (1.4)

(See [19, Equation (1.2)].) When $\deg(P)=2$, this conjecture was verified by
Helfgott [18], based on ideas of de la Vallée-Poussin [3, 4]. Helfgott later
extended his analysis to cover the case $\deg(P)=3$ in [19]. More recently,
Green and Tao [15] established (1.4) when $P$ is a product of pairwise
independent linear forms.

In place of the Liouville function $\lambda$, a similar question can be asked
about an arbitrary completely multiplicative function. In this direction,
Frantzikinakis and Host [12] established the analogous statement to (1.4) for
any “aperiodic” multiplicative function and for a class of polynomials $P$
which includes all products of pairwise independent linear forms. They later
posed the following question, which was a major motivator for our current
paper.

######  Question 1.1 ([11, Page 91]).

Let $f:\mathbb{N}\to\mathbb{R}$ be a real valued bounded completely
multiplicative function and let $P\in\mathbb{Z}[x,y]$ be a homogeneous
polynomial with values on the positive integers. Does the limit

$\lim_{N\to\infty}\frac{1}{N^{2}}\sum_{m,n=1}^{N}f(P(m,n))$

exist?

Already in [11], Frantzikinakis and Host provided a positive answer to 1.1 in
the special case when the polynomial $P$ is a product of linear forms. Shortly
after, Klurman and Mangerel [20] obtained a concrete, effective formula for
the averages in that case. A related work was also done by Matthiesen in [24].
Nevertheless, when $P$ is not a product of linear forms, the answer to 1.1
remains elusive, and even solving it for specific functions $f$ poses
significant challenges.

Our first main result answers 1.1 for the polynomial $P(m,n)=m^{2}+n^{2}$ and
for an arbitrary completely multiplicative function $f$. Given a function
$f:\\{1,\ldots,N\\}\to\mathbb{C}$, we write $\mathop{\mathbb{E}}_{1\leq
m,n\leq N}f(x)$ for the average $\frac{1}{N^{2}}\sum_{1\leq m,n\leq N}f(x)$.
More generally, if $A$ is a finite set and $f:A\to\mathbb{C}$ is a function on
$A$, we use $\mathop{\mathbb{E}}_{x\in A}f(x)$ as a shorthand notation for
$\frac{1}{|A|}\sum_{x\in A}f(x)$.

######  Theorem A.

Let $f:\mathbb{N}\to\mathbb{R}$ be a bounded completely multiplicative
function. Then the average

$\lim_{N\to\infty}\mathop{\mathbb{E}}_{1\leq m,n\leq N}f(m^{2}+n^{2})$

exists and equals

$\frac{1}{2-f(2)}\cdot\prod_{\begin{subarray}{c}p\text{ prime }\\\ p\equiv
1\bmod
4\end{subarray}}\left(\frac{p-1}{p-f(p)}\right)^{2}\cdot\prod_{\begin{subarray}{c}p\text{
prime }\\\ p\equiv 3\bmod 4\end{subarray}}\frac{p^{2}-1}{p^{2}-f(p)^{2}}.$
(1.5)

A main tool used in the proof of Frantzikinakis and Host’s result [11]
mentioned above is a structure theorem of multiplicative functions developed
in [12] which roughly speaking says that any bounded multiplicative function
can be decomposed into a component that resembles a periodic function and a
Gowers–uniform error term (see also [28, 29]). Although this structure theorem
provides an effective way to deal with multiplicative functions along linear
forms, it does not seem to help for general higher degree polynomials. An
alternative approach to handle certain higher degree polynomials is to realize
them as norm forms of number fields. In this paper we focus on the polynomial
$P(m,n)=m^{2}+n^{2}$, which can naturally be viewed as the norm function over
the Gaussian integers $\mathbb{G}:=\\{m+\mathrm{i}n:m,n\in\mathbb{N}\\}$.
Recall that the norm function over $\mathbb{G}$ is
$\mathcal{N}(m+\mathrm{i}n)=m^{2}+n^{2}$ and it satisfies
$\mathcal{N}(ab)=\mathcal{N}(a)\mathcal{N}(b)$ for any $a,b\in\mathbb{G}$.
Therefore, given a completely multiplicative function
$f:\mathbb{N}\to\mathbb{R}$, the composition $f\circ\mathcal{N}$ is a
completely multiplicative function from the set of non-zero Gaussian integers
$\mathbb{G}^{*}$ to $\mathbb{R}$. Using this observation, we are able to
derive A from a version of Wirsing’s theorem for Gaussian integers.

######  Theorem 1.2.

If $f:\mathbb{G}^{*}\to\mathbb{R}$ is a real-valued bounded completely
multiplicative function, then the limit

$\lim_{N\to\infty}\mathop{\mathbb{E}}_{1\leq m,n\leq N}f(m+\mathrm{i}n)$

exists.

It is possible to identify the limit in Theorem 1.2 as an Euler product. To
write this product, denote by $\mathbb{P}$ the set of Gaussian primes and by
$\mathbb{P}_{1}$ the set of Gaussian primes in the first quadrant. (We discuss
basic properties of Gaussian primes in Section 2). Given a bounded completely
multiplicative function $f:\mathbb{G}^{*}\to\mathbb{C}$, we define $P(f)$ as

$P(f)\coloneqq\prod_{p\in\mathbb{P}_{1}}\frac{\mathcal{N}(p)-1}{\mathcal{N}(p)-f(p)}.$
(1.6)

We remark that the infinite product defining $P(f)$ does not necessarily
converge for every bounded completely multiplicative function
$f:\mathbb{G}^{*}\to\mathbb{C}$; however, it does when $f$ takes real values
(see Lemma 4.5).

At this stage we move from averages over squares to the more general situation
of averages over what we call _dilated Følner sequences_. This concept will be
introduced and discussed in detail in Section 2.3; for now, it suffices to say
that examples of dilated Følner sequences include the sequence of squares
$\Phi_{N}=\\{m+\mathrm{i}n\in\mathbb{G}^{*}:1\leq m,n\leq N\\}$ and the
sequence of discs
$\Phi_{N}=\\{m+\mathrm{i}n\in\mathbb{G}^{*}:0<m^{2}+n^{2}<N^{2}\\}$. We can
now formulate a stronger version of Theorem 1.2.

######  Theorem B.

If $f:\mathbb{G}^{*}\to\mathbb{R}$ is a real-valued bounded completely
multiplicative function satisfying $f(\mathrm{i})=1$ and
$(\Phi_{N})_{N\in\mathbb{N}}$ is a dilated Følner sequence, then
$\lim_{N\to\infty}\mathop{\mathbb{E}}_{n\in\Phi_{N}}f(n)$ exists and equals
$P(f)$.

######  Remark 1.3.

The assumption $f(\mathrm{i})=1$ is important in B: for instance if $\Phi$ is
a finite set invariant under multiplication by $\mathrm{i}$ (such as a
centered square or a disc), the average $\mathop{\mathbb{E}}_{n\in\Phi}f(n)$
equals $0$ whenever $f(\mathrm{i})\neq 1$. However, the condition
$f(\mathrm{i})=1$ is not necessary for the limit to exist. In Theorem 4.7
below we derive from B a more general version without the assumption that
$f(\mathrm{i})=1$.

######  Remark 1.4.

B is false for general (i.e., “non-dilated”) Følner sequences. More precisely,
in Section 2.3, we show that there exists a completely multiplicative function
$f:\mathbb{G}^{*}\to\\{-1,1\\}$ and an additive Følner sequence
$(\Phi_{N})_{N\in\mathbb{N}}$ such that
$\lim_{N\to\infty}\mathop{\mathbb{E}}_{n\in\Phi_{N}}f(n)$ does not exist.
Furthermore, a modification of an argument by Fish [10] shows that if one
considers a random completely multiplicative function
$f:\mathbb{G}^{*}\to\\{-1,1\\}$, then almost surely, $f$ will contain all
finite patterns of $-1$ and $1$. Details are given in Appendix B.

Halász’s theorem mentioned above, which contains Wirsing’s theorem as a
special case, was extended to arbitrary function fields by Granville, Harper,
and Soundararajan [14]. On the other hand, as far as we know, there is no
analogue of Wirsing’s and Halász’s theorems in the number field setting. B
partially fills this gap because it can be seen as an analogue of Wirsing’s
theorem for completely multiplicative functions on the Gaussian integers. In
fact, we prove a more general result, which partially explains why the
restriction that $f$ takes on real values is convenient in Wirsing’s theorem.

For $z\in\mathbb{C}\setminus\\{0\\}$, let $\operatorname{Arg}(z)\in[-\pi,\pi)$
denote its argument, and define $\operatorname{Arg}(0)=0$. Any real number $x$
has $\operatorname{Arg}(x)=0$ or $-\pi$, so B is a special case of the
following theorem.

######  Theorem C.

Let $f:\mathbb{G}^{*}\to\mathbb{C}$ be a bounded completely multiplicative
function such that $f(\mathrm{i})=1$ and $\operatorname{Arg}(f(\mathbb{P}))$
is finite. Then for every dilated Følner sequence
$(\Phi_{N})_{N\in\mathbb{N}}$, the limit
$\lim_{N\to\infty}\mathop{\mathbb{E}}_{n\in\Phi_{N}}f(n)$ exists and equals
$P(f)$.

Similarly to Remark 1.3, it is possible to remove from C the assumption that
$f(\mathrm{i})=1$ (see Theorem 4.7).

###  1.2. Ergodic averages along $\Omega(m^{2}+n^{2})$ in uniquely ergodic
systems

To prove Theorems B and C, instead of following the more classical approach of
Wirsing, Delange, and Halász, we opt to use a dynamical approach, as developed
in the recent work of Bergelson and Richter [1]. Recall that, for a natural
number $n\in\mathbb{N}$, $\Omega(n)$ is the number of prime factors of $n$
counted with multiplicities. A _uniquely ergodic system_ is a pair $(X,T)$
where $X$ is a compact metric space, $T:X\to X$ is a continuous map and there
exists a unique Borel probability measure $\mu$ on $X$ satisfying
$\mu(T^{-1}A)=\mu(A)$ for every Borel set $A\subset X$. The following result
was proved in [1].

######  Theorem 1.5 ([1, Theorem A]).

Let $(X,T)$ be a uniquely ergodic system with the unique invariant measure
$\mu$. For any $x\in X$ and any continuous function $f:X\to\mathbb{C}$,

$\lim_{N\to\infty}\mathop{\mathbb{E}}_{1\leq n\leq
N}f(T^{\Omega(n)}x)=\int_{X}f\ d\mu.$

Theorem 1.5, when applied to the two-points system, reduces to (an equivalent
form of) the prime number theorem. Applying it to finite systems, one recovers
a theorem of Pillai [25] and Selberg [27] stating that $\Omega(n)$ is equally
distributed over all residue classes mod $q$ for all $q\in\mathbb{N}$. Theorem
1.5 also contains as a special case (when applying it to irrational rotations
on the torus) the Erdős-Delange Theorem [5], which complements the results
above by stating that for any irrational number $\alpha$, the sequence
$(\Omega(n)\alpha)_{n\in\mathbb{N}}$ is uniformly distributed mod $1$. Here, a
sequence $(a(n))_{n\in\mathbb{N}}$ of real numbers is called _uniformly
distributed mod $1$_ if for every interval $I\subset[0,1)$,

$\lim_{N\to\infty}\mathop{\mathbb{E}}_{1\leq n\leq N}1_{I}(a(n)\bmod 1)=|I|.$

Our next theorem is an analogue of Theorem 1.5 along sums of two squares.

######  Theorem D.

Let $(X,T)$ be a uniquely ergodic system with the unique invariant measure
$\mu$. Then for any $x\in X$ and any $f\in C(X)$,

$\lim_{N\to\infty}\mathop{\mathbb{E}}_{1\leq m,n\leq
N}f(T^{\Omega(m^{2}+n^{2})}x)=\int_{X}f\ d\mu.$

A sequence $(a(m,n))_{m,n\in\mathbb{N}}$ of two parameters is called
_uniformly distributed mod $1$_ if for every interval $I\subset[0,1)$,

$\lim_{N\to\infty}\mathop{\mathbb{E}}_{1\leq m,n\leq N}1_{I}(a(m,n)\bmod
1)=|I|.$

Taking $(X,T)$ to be the rotation by $q$ points, D implies that
$\Omega(m^{2}+n^{2})$ is equally distributed over all residue classes mod $q$.
When applied to the rotation by an irrational $\alpha$ on the torus
$\mathbb{R}/\mathbb{Z}$, we have $(\Omega(m^{2}+n^{2})\alpha)$ is uniformly
distributed mod $1$. By applying D to unipotent affine transformations on tori
(following the approach of Furstenberg in [13, pages 67 – 69]), we obtain the
following corollary:

######  Corollary 1.6.

For $Q\in\mathbb{R}[x]$, $Q(\Omega(m^{2}+n^{2}))_{m,n\in\mathbb{N}}$ is
uniformly distributed mod $1$ if and only if at least one of the coefficients
of $Q$ is irrational.

###  1.3. The main theorem

The main technical result of this paper, stated below, is an ergodic theorem
involving additive averages for multiplicative actions of the Gaussian
integers. This theorem, albeit somewhat complicated to formulate, is the main
ingredient in the proofs of Theorems C and D.

######  Theorem E.

Let $X$ be a compact metric space and let $\mathcal{T}$ denote the semigroup
of all continuous transformations $T:X\to X$ under composition. Let
$\tau:(\mathbb{G}^{*},\times)\to\mathcal{T}$ be a semigroup homomorphism such
that $\tau(\mathbb{P})$ is finite and let
$T_{1},\dots,T_{k}\in\tau(\mathbb{P})$ be the complete list of those
transformations satisfying

$\sum_{p\in\mathbb{P}:\tau(p)=T_{j}}\frac{1}{\mathcal{N}(p)}=\infty.$ (1.7)

Suppose that there exists a unique Borel probability measure $\mu$ on $X$ such
that for all $j\in[k]$ and every Borel set $A\subset X$,
$\mu(T_{j}^{-1}A)=\mu(A)$. Then for any $x\in X$, $F\in C(X)$ and dilated
Følner sequence $(\Phi_{N})_{N\in\mathbb{N}}$,

$\lim_{N\to\infty}\mathop{\mathbb{E}}_{n\in\Phi_{N}}F(\tau(n)x)=\int_{X}F\
d\mu.$

E is a direct analogue of [1, Theorem B], which was formulated for “finitely
generated, strongly uniquely ergodic, multiplicative dynamical systems”. While
drawing inspiration from [1, Theorem B], the proof of E contains some major
differences due to the more intricate geometry inherent in $\mathbb{G}$, a
rank two additive group, in contrast to the simpler rank one group,
$\mathbb{Z}$. Moreover, we must handle arbitrary dilated Følner sequences in
$\mathbb{G}^{*}$, which are not as rigid as the sequence of intervals
$\\{1,\dots,N\\}$ considered in [1] (see Remark 2.2).

We also emphasize that the proof of E is dynamical and combinatorial in
nature, and the only number theoretic input needed (besides a version of the
Turán-Kubilius inequality) is some control on the distribution of the primes
in $\mathbb{G}$, which already follows from the works of Landau [21] and Hecke
[17].

E involves taking additive averages on a multiplicative dynamical system and
is reminiscent of the results in our previous paper [7] regarding measure
preserving actions of $(\mathbb{N},\times)$. To put this in perspective, as
opposed to the main results in [7], which dealt with arbitrary multiplicative
actions, in E we require the assumption that $\tau(\mathbb{P})$ is finite. On
the other hand, the conclusion in E is significantly stronger than the results
in [7].

###  1.4. Outline of the article

In Section 2, we set up notation and present some basic facts about Gaussian
integers, the prime number theorem in $\mathbb{G}$, and define dilated Følner
sequences. The proof of E occupies Section 3. A short proof of D is given in
Section 4.4 and C is proved in Section 4.2. Some natural open questions are
listed in Section 5. Lastly, the appendix contains some technical estimates
which are needed for the proof of E and an argument showing that a random
multiplicative functions taking values in $\\{-1,1\\}$ almost surely contains
all finite patterns of $-1$ and $1$.

Acknowledgements. The authors would like to thank Vitaly Bergelson and Florian
Richter for helpful discussions about their paper [1].

##  2\. Background

###  2.1. Notation

The absolute value of a complex number $z$ is written $|z|$, and its argument
is denoted by $\operatorname{Arg}(z)\in\mathbb{R}/(2\pi\mathbb{Z})$, usually
identified with $[-\pi,\pi)$. We also use the convention that
$\operatorname{Arg}(0)=0$. As usual, we denote by $\mathrm{Re}\,z$ and
$\mathrm{Im}\,z$ the real and imaginary parts of $z\in\mathbb{C}$. We let
$S^{1}:=\\{z\in\mathbb{C}\colon|z|=1\\}\subset\mathbb{C}$ be the unit circle
and $\mathbb{D}:=\\{z\in\mathbb{C}:|z|\leq 1\\}$ be the unit disk.

We denote by $\mathbb{G}$ the ring of Gaussian integers
$\\{a+b\mathrm{i}:a,b\in\mathbb{Z}\\}\subset\mathbb{C}$, and we use
$\mathbb{G}^{\ast}:=\mathbb{G}\backslash\\{0\\}$ for the set of non-zero
Gaussian integers. The norm of a Gaussian integer $n$ is defined by
$\mathcal{N}(n)=|n|^{2}$.

The greatest common divisor of two non-zero Gaussian integers $m,n$ is only
unique up to multiplication by a unit, so we avoid using $\gcd(m,n)$ on its
own; however the norm $\mathcal{N}(\gcd(m,n))$ is well defined.

Given $A\subset\mathbb{C}$ and $z\in\mathbb{C}$, we define $A\pm z=\\{a\pm
z:a\in A\\}$, and $zA=\\{za:a\in A\\}$. Depending on $A$ and $z$, we assign
different meanings to $A/z$: if $A\subset\mathbb{C}$ and
$z\in\mathbb{C}\setminus\\{0\\}$, define $A/z=\\{a/z:z\in A\\}$ _unless_
$A\subset\mathbb{G}$ and $z\in\mathbb{G}^{*}$, in which case we define
$A/z=\\{x\in\mathbb{G}:xz\in A\\}$. This distinction will be clear from the
context.

When $A$ is a finite set, we denote by $|A|$ its cardinality. Given a subset
$A\subset\mathbb{G}^{\ast}$, denote by

$\log(A):=\sum_{n\in A}\frac{1}{\mathcal{N}(n)}$

its _logarithmic weight_. If $\log(A)=\infty$ we say that $A$ is a _divergent
set_. Otherwise we say that $A$ is a _convergent set_. It is clear that every
divergent set is infinite, and whenever a divergent set is partitioned into
finitely many sets, at least one of them must be divergent.

For a non-empty finite set $A$ and function $f:A\to\mathbb{C}$, we set:

$\mathop{\mathbb{E}}_{n\in A}f(n)\coloneqq\frac{1}{|A|}\sum_{n\in A}f(n).$

If $A\subset\mathbb{G}^{*}$, then we set:

$\mathop{\mathbb{E}}^{\log}_{n\in A}f(n)\coloneqq\frac{1}{\log(A)}\sum_{n\in
A}\frac{f(n)}{\mathcal{N}(n)}.$

Given two functions $f,g:\mathbb{G}^{\ast}\to\mathbb{C}$, we use the following
notations:

  * •

$\displaystyle f(x)=O(g(x))$ or $\displaystyle f(x)\ll g(x)$ means there is a
positive constant $C$ such that $|f(x)|<Cg(x)$ for all
$x\in\mathbb{G}^{\ast}$.

  * •

$\displaystyle f(x)=o(g(x))$ indicates that

$\lim_{\mathcal{N}(x)\to\infty}\frac{f(x)}{g(x)}=0.$

  * •

$\displaystyle f(x)\sim g(x)$ means

$\lim_{\mathcal{N}(x)\to\infty}\frac{f(x)}{g(x)}=1.$

###  2.2. Distribution of Gaussian primes

A Gaussian prime is an element $p\in\mathbb{G}^{*}$ which cannot be decomposed
as $p=ab$ where $a,b$ are non-unit Gaussian integers. We use $\mathbb{P}$ to
denote the set of Gaussian primes and
$\mathbb{P}_{1}=\\{p\in\mathbb{P}:\mathrm{Re}\,p>0,\mathrm{Im}\,p\geq 0\\}$
for the restriction of $\mathbb{P}$ to the first quadrant. A Gaussian integer
$a+b\mathrm{i}$ in the first quadrant is a Gaussian prime if and only if
either:

  * •

$b=0$ and $a$ is a prime in $\mathbb{N}$ of the form $4n+3$, or

  * •

$b>0$ and $a^{2}+b^{2}$ is a prime number in $\mathbb{N}$ (which will not be
of the form $4n+3)$.

Note that the units of $\mathbb{G}$ are $\\{1,\mathrm{i},-1,-\mathrm{i}\\}$
and hence the first quadrant is a natural fundamental domain for their action
on $\mathbb{G}$. In particular, $\mathbb{P}$ is invariant under multiplication
by $\mathrm{i}$.

In our proofs we make crucial use of known results about the distribution
$\mathbb{P}$. In analogy to the prime number theorem, Landau [21] proved that

$\big{|}\\{p\in\mathbb{P}:\mathcal{N}(p)<N\\}\big{|}\sim\frac{N}{\log N}\text{
as }N\to\infty.$ (2.1)

Landau’s result only estimates the number of primes in a disk around the
origin. This was later extended by Hecke [17] who showed that the number of
primes in a “slice” (or sector) of the complex plane is proportional to its
amplitude.

######  Theorem 2.1 ([17], see also [26]).

For any interval $I\subset[0,\pi/2]$,

$\frac{|\\{p\in\mathbb{P}:\mathcal{N}(p)<N\text{ and }\operatorname{Arg}(p)\in
I\\}|}{N/\log N}\to\frac{|I|}{2\pi}\text{ as }N\to\infty.$

As a corollary of Theorem 2.1 it follows that for any interval
$I\subset[-\pi,\pi)$ and any $0\leq a<b$,

$\frac{|\\{p\in\mathbb{P}:aN\leq\mathcal{N}(p)\leq bN\text{ and
}\operatorname{Arg}(p)\in I\\}|}{N/\log N}\to\frac{|I|}{2\pi}(b-a)\text{ as
}N\to\infty.$ (2.2)

We will use (2.2) to estimate the amount of primes in dilations of certain
small neighborhoods of $1$. While we could use disks centered at $1$, in view
of (2.2) it is more convenient to use the following annulus sectors: For each
$\varepsilon\in(0,1)$ define

$S_{\varepsilon}:=\big{\\{}z\in\mathbb{C}:1-\varepsilon<|z|<1+\varepsilon,\
\operatorname{Arg}(z)\in\left(-\pi\varepsilon,\pi\varepsilon\right)\big{\\}}.$
(2.3)

For each $\varepsilon\in(0,1)$ and $n\in\mathbb{G}^{*}$, since
$\mathcal{N}(n)=|n|^{2}$, we have

$nS_{\varepsilon}\cap\mathbb{G}=\Big{\\{}m\in\mathbb{G}:(1-\varepsilon)^{2}\mathcal{N}(n)<\mathcal{N}(m)<(1+\varepsilon)^{2}\mathcal{N}(n),\
\operatorname{Arg}(m)\in\operatorname{Arg}(n)+\big{(}-\pi\varepsilon,\pi\varepsilon\big{)}\Big{\\}}.$

Using (2.2), we deduce that for every $\varepsilon>0$,

$\lim_{n\in\mathbb{G}^{\ast},\mathcal{N}(n)\to\infty}\frac{\big{|}\mathbb{P}\cap
nS_{\varepsilon}\big{|}}{\mathcal{N}(n)/\log\mathcal{N}(n)}=\frac{2\pi\varepsilon(4\varepsilon)}{2\pi}=4\varepsilon^{2}.$
(2.4)

###  2.3. Dilated Følner sequences

Given a function $f:\mathbb{Z}\to\mathbb{C}$ it is natural to consider its
Cesàro average $\mathop{\mathbb{E}}_{1\leq n\leq N}f(n)$ over the initial
interval $\\{1,\dots,N\\}$, or the average $\mathop{\mathbb{E}}_{-N\leq n\leq
N}f(n)$ over the centered interval $\\{-N,\dots,N\\}$. However, when given a
function $f:\mathbb{G}\to\mathbb{C}$, there are several natural ways to
average $f$: one can consider, for instance, averages over disks
$\mathop{\mathbb{E}}_{\mathcal{N}(n)<N}f(n)$ or over the squares
$\mathop{\mathbb{E}}_{1\leq m,n\leq N}f(m+n\mathrm{i})$ or
$\mathop{\mathbb{E}}_{-N\leq m,n\leq N}f(m+n\mathrm{i})$. To simultaneously
address all these averaging schemes, we make use of the notion of (additive)
_Følner sequences_ ; these are sequences $\Phi=(\Phi_{N})_{N\in\mathbb{N}}$ of
finite subsets of $\mathbb{G}$ such that for every $n\in\mathbb{G}$,

$\lim_{N\to\infty}\frac{\big{|}(\Phi_{N}+n)\triangle\Phi_{N}\big{|}}{|\Phi_{N}|}=0.$

######  Remark 2.2.

Another reason we work with abstract Følner sequences, as opposed to simply
use the squares
$\Phi_{N}:=\\{n\in\mathbb{G}^{*}:0<\mathrm{Re}\,n,\mathrm{Im}\,n\leq N\\}$ is
that we often have to consider the average over the set $\Phi_{N}/a$, for some
$a\in\mathbb{G}$. Unlike the situation in $\mathbb{Z}$, the set $\Phi_{N}/a$
does not necessarily equal $\Phi_{M}$ for some other $M$. Therefore, even if
one is interested solely in averages over squares, it is necessary to consider
more general Følner sequences.

As already mentioned in the introduction, not every Følner sequence works in
our results. Indeed, by [10, Theorem 1.1], there is a real-valued completely
multiplicative function $g\colon\mathbb{N}\to\\{-1,1\\}$ such that both
$g^{-1}(\\{1\\})$ and $g^{-1}(\\{-1\\})$ contain arbitrarily long intervals.
Consider $f\colon\mathbb{G}^{*}\to\\{-1,1\\}$ defined as
$f(n)=g(\mathcal{N}(n))$ for $n\in\mathbb{G}^{*}$. It follows that
$f^{-1}(\\{1\\})$ and $f^{-1}(\\{-1\\})$ contain Følner sequences
$(\Phi^{+}_{N})_{N\in\mathbb{N}}$ and $(\Phi^{-}_{N})_{N\in\mathbb{N}}$ in
$\mathbb{G}$, respectively. Letting $\Phi_{2N}=\Phi^{+}_{N}$ and
$\Phi_{2N+1}=\Phi^{-}_{N}$, we see $\mathop{\mathbb{E}}_{n\in\Phi_{N}}f(n)$
does not converge.

There are plenty of functions $f$ as above. One can extend Fish’s theorem [10]
on random multiplicative functions from $\mathbb{Z}^{\ast}$ to
$\mathbb{G}^{\ast}$ to show that most completely multiplicative functions
$f:\mathbb{G}^{\ast}\to\\{-1,1\\}$ contains all finite patterns of $-1$ and
$1$. This is a result of independent interest, but because it is somewhat
distinct from the primary content of the article, we put it in Appendix B.

Due to the above reason, in Theorems B, C, and E, we must restrict to a
subclass of Følner sequences obtained from dilating an open set.

######  Definition 2.3.

A sequence $(\Phi_{N})_{N\in\mathbb{N}}$ of finite subsets of $\mathbb{G}^{*}$
is called a _dilated Følner sequence_ if there exists a Jordan measurable111A
set $U\subset\mathbb{C}$ is Jordan measurable if and only if it is bounded and
its boundary has measure zero. This is equivalent to the indicator function
$1_{U}$ being Riemann integrable. set $U\subset\mathbb{C}$ and a sequence of
positive real numbers $(k_{N})_{N\in\mathbb{N}}$ such that $k_{N}\to\infty$ as
$N\to\infty$ and $\Phi_{N}=\mathbb{G}^{\ast}\cap k_{N}U$.

We stress that the definition does not require $U$ to contain $0$ and it is
easy to check that every dilated Følner sequence is indeed an additive Følner
sequence.

######  Example 2.4.

  1. (i)

Two natural dilated Følner sequences are the squares
$\Phi_{N}:=\\{n\in\mathbb{G}^{*}:0<\mathrm{Re}\,n,\mathrm{Im}\,n\leq N\\}$ and
the disks $\Phi_{N}=\\{z\in\mathbb{G}^{*}:\mathcal{N}(z)\leq N^{2}\\}$.

  2. (ii)

It is immediate from the definition that any subsequence of a dilated Følner
sequence is a dilated Følner sequence.

  3. (iii)

If $(\Phi_{N})_{N\in\mathbb{N}}$ is a dilated Følner sequence in $\mathbb{G}$
and $n\in\mathbb{G}$, then the sets $\Phi_{N}=\Phi_{N}/n$ also form a dilated
Følner sequence.

  4. (iv)

The sequence of shifted disks $\Phi_{N}=N^{2}+(N\mathbb{D}\cap\mathbb{G})$ is
not a dilated Følner sequence.

It remains an interesting question whether the Følner sequence in part (iv) of
the example satisfies the conclusion of E (see also the related 5.5).

The main property that distinguishes dilated Følner sequences is captured in
the following lemma and, roughly speaking, states that dilated Følner
sequences are unchanged under small multiplicative perturbations.

######  Lemma 2.5.

If $(\Phi_{N})_{N\in\mathbb{N}}$ is a dilated Følner sequence, then for every
$\delta>0$, there exists $\varepsilon>0$ such that whenever
$a,b\in\mathbb{G}^{\ast}$ satisfy $b\in aS_{\varepsilon}$, then

$\lim_{N\to\infty}\frac{|\Phi_{N}/a\triangle\Phi_{N}/b|}{|\Phi_{N}/a|}<\delta.$

###### Proof.

Denote by $m$ the Lebesgue measure on $\mathbb{C}$. If $V\subset\mathbb{C}$ is
a Jordan measurable set and $t$ denotes a real parameter, then
$|\mathbb{G}^{*}\cap tV|/t^{2}\to m(V)$ as $t\to\infty$. Indeed,

$\lim_{t\to\infty}\frac{|\mathbb{G}^{*}\cap
tV|}{t^{2}}=\lim_{t\to\infty}\frac{1}{t^{2}}\sum_{n\in
G^{*}}1_{V}(n/t)=\lim_{\varepsilon\to 0}\varepsilon^{2}\sum_{n\in
G^{*}}1_{V}(n\varepsilon).$

The last sum is a Riemann sum for $1_{V}$ and since $1_{V}$ is Riemann
integrable, the limit exists and equals
$\int_{\mathbb{C}}1_{V}\,\mathrm{d}m=m(V)$.

Now let $U\subset\mathbb{C}$ be a Jordan measurable set and $(k_{N})$ be a
sequence of real numbers such that $\Phi_{N}=k_{N}U\cap\mathbb{G}^{*}$. Then
$\Phi_{N}/a=\\{n\in\mathbb{G}^{*}:n\in k_{N}U/a\\}=k_{N}\tfrac{U}{a}\cap
G^{*}$. Since the set $\tfrac{U}{a}$ is Jordan measurable, it follows that
$|\Phi_{N}/a|/k_{N}^{2}\to m(\tfrac{U}{a})=m(U)/\mathcal{N}(a)$. Similarly,
$\Phi_{N}/a\cap\Phi_{N}/b=k_{N}\Big{(}\tfrac{U}{a}\cap\tfrac{U}{b}\Big{)}\cap
G^{*}$, so $|\Phi_{N}/a\cap\Phi_{N}/b|/k_{N}^{2}\to
m\Big{(}\tfrac{U}{a}\cap\tfrac{U}{b}\Big{)}=m\Big{(}U\cap\tfrac{a}{b}U\Big{)}/\mathcal{N}(a)$.
It follows that

$\lim_{N\to\infty}\frac{|\Phi_{N}/a\cap\Phi_{N}/b|}{|\Phi_{N}/a|}=\frac{m(U\cap(a/b)U)}{m(U)}=1-o_{a/b\to
1}(1).$

∎

We use Lemma 2.5 only to prove Lemma 3.7.

##  3\. Proof of E

In this section we prove E. Throughout the section we fix a compact metric
space $X$, we let ${\mathcal{T}}$ denote the semigroup of continuous functions
$T:X\to X$, and we fix a semigroup homomorphism
$\tau:(\mathbb{G},\times)\to{\mathcal{T}}$ such that $\tau(\mathbb{P})$ is
finite. We also fix the transformations $T_{1},\dots,T_{k}\in{\mathcal{T}}$
such that each set $\mathbb{P}\cap\tau^{-1}(T_{i})$ is divergent and the
remainder set $\\{p\in\mathbb{P}:\tau(p)\notin\\{T_{1},\dots,T_{k}\\}\\}$ is
convergent.

We say that a Borel probability measure $\nu$ on $X$ is an _additively
empirical measure_ if there exists $x\in X$ and some dilated Følner sequence
$\Phi$ such that $\nu$ is a weak∗ limit

$\nu=\lim_{N\to\infty}\mathop{\mathbb{E}}_{n\in\Phi_{N}}\tau(n)\delta_{x},$

where $\delta_{x}$ denotes the Dirac measure at the $x$, and
$\tau(n)\delta_{x}$ denotes the pushforward of $\delta_{x}$ under the
transformation $\tau(n)\in{\mathcal{T}}$.

The first step of the proof of E is to reduce it to the following statement,
whose proof occupies most of the section.

######  Theorem 3.1.

For every additively empirical measure $\nu$ and every $j\in[k]$,
$T_{j}\nu=T_{j}^{2}\nu$.

###### Proof of E assuming Theorem 3.1.

Let $\mu$ be the unique probability measure on $X$ invariant under all the
maps $T_{j}$ for $j\in[k]$. We need to show that for each $x\in X$, $f\in
C(X)$ and every dilated Følner sequence $(\Phi_{N})_{N\in\mathbb{N}}$,

$\lim_{N\to\infty}\mathop{\mathbb{E}}_{n\in\Phi_{N}}f(\tau(n)x)=\int_{X}F\,\mathrm{d}\mu.$

This is equivalent to the statement that $\mu$ is the unique additively
empirical measure. Since $\mu$ is the unique probability measure on $X$
invariant under each $T_{j}$, our task will be complete if we show that any
additively empirical measure $\nu$ on $X$ satisfies $T_{j}\nu=\nu$ for every
$j\in[k]$. This would follow directly from Theorem 3.1 if each $T_{j}$ were
invertible, but that is not necessarily true. Nevertheless, it is still
possible to apply Theorem 3.1 and the special nature of additively empirical
measures to conclude that $T_{j}\nu=\nu$ for every $j\in[k]$.

Fix $j\in[k]$. Let $x\in X$ and let $\Phi=(\Phi_{N})_{N\in\mathbb{N}}$ be a
dilated Følner sequence such that
$\nu=\lim_{N\to\infty}\mathop{\mathbb{E}}_{n\in\Phi_{N}}\tau(n)\delta_{x}$.
Fix $f\in C(X)$ and $\varepsilon>0$. We will use a version of the Turán-
Kubilius inequality for Gaussian integers to compare the averages of
$f(\tau(n)x)$ with the averages of $f(T_{j}\tau(n)x)$; for completeness we
formulate and prove the version we need in Appendix A.

Let $B\subset\mathbb{P}\cap\tau^{-1}(T_{j})$ be such that
$\log(B)>4/\varepsilon^{2}$ and apply Lemma A.1 with $a(n)=f(\tau(n)x)$
together with Lemma A.2 to conclude that

$\limsup_{N\to\infty}\left|\mathop{\mathbb{E}}_{n\in\Phi_{N}}f(\tau(n)x)-\mathop{\mathbb{E}}^{\log}_{p\in
B}\mathop{\mathbb{E}}_{n\in\Phi_{N}/p}f(T_{j}\tau(n)x)\right|<\varepsilon.$
(3.1)

On the other hand, applying Lemma A.1 with $a(n)=f(T_{j}\tau(n)x)$ together
with Lemma A.2, it follows that

$\limsup_{N\to\infty}\left|\mathop{\mathbb{E}}_{n\in\Phi_{N}}f(T_{j}\tau(n)x)-\mathop{\mathbb{E}}^{\log}_{p\in
B}\mathop{\mathbb{E}}_{n\in\Phi_{N}/p}f(T_{j}^{2}\tau(n)x)\right|<\varepsilon.$
(3.2)

Note that, for every $p\in\mathbb{P}$, the sequence
$(\Phi_{N}/p)_{N\in\mathbb{N}}$ is a dilated Følner sequence, and hence any
accumulation point of the measures
$\mathop{\mathbb{E}}_{n\in\Phi_{N}/p}\tau(n)\delta_{x}$ is an additively
empirical measure $\tilde{\nu}$. Using Theorem 3.1 we deduce that
$T_{j}\tilde{\nu}=T_{j}^{2}\tilde{\nu}$, which implies that

$\forall
p\in\mathbb{P},\qquad\lim_{N\to\infty}\left|\mathop{\mathbb{E}}_{n\in\Phi_{N}/p}f(T_{j}\tau(n)x)-\mathop{\mathbb{E}}_{n\in\Phi_{N}/p}f(T_{j}^{2}\tau(n)x)\right|=0.$

Combining this with (3.1) and (3.2), we conclude that

$\limsup_{N\to\infty}\left|\mathop{\mathbb{E}}_{n\in\Phi_{N}}f(\tau(n)x)-\mathop{\mathbb{E}}_{n\in\Phi_{N}}f(T_{j}\tau(n)x)\right|<2\varepsilon.$

Taking $\varepsilon\to 0$ it follows that

$\int_{X}f\,\mathrm{d}\nu=\lim_{N\to\infty}\mathop{\mathbb{E}}_{n\in\Phi_{N}}f(\tau(n)x)=\lim_{N\to\infty}\mathop{\mathbb{E}}_{n\in\Phi_{N}}f(T_{j}\tau(n)x)=\int_{X}f\,\mathrm{d}(T_{j}\nu).$

Since $f$ was arbitrary we conclude that $\nu=T_{j}\nu$, and this finishes the
proof. ∎

###  3.1. Roadmap of the proof of Theorem 3.1

Our main (and only) tool to show that $T\nu=R\nu$ for some transformations
$T,R\in{\mathcal{T}}$ is the following result which is inspired by [1,
Proposition 2.1].

######  Lemma 3.2.

Suppose that $T,R\in{\mathcal{T}}$ commute and for every $\varepsilon>0$ there
are a finite set $F\subset\mathbb{G}^{\ast}$ and an injective map
$\alpha:F\to\mathbb{G}^{\ast}$ such that

  * •

$\alpha(n)\in nS_{\varepsilon}$ for all $n\in F$, where $S_{\varepsilon}$ is
defined in (2.3),

  * •

$\tau(F)=\\{T\\}$ and $\tau(\alpha(F))=\\{R\\}$.

  * •

$\displaystyle\mathop{\mathbb{E}}_{n,m\in
F}^{\log}\mathcal{N}(\gcd(n,m))<1+\varepsilon$ and
$\displaystyle\mathop{\mathbb{E}}_{n,m\in\alpha(F)}^{\log}\mathcal{N}(\gcd(n,m))<1+\varepsilon$.

Then $T\nu=R\nu$ for every empirical measure $\nu$.

Lemma 3.2 is proved, using a variant of the Turán-Kubilius inequality, in
Section 3.2 below.

Unfortunately, Lemma 3.2 requires strong assumptions, and in particular we can
not use it directly with $(T,R)=(T_{j},T_{j}^{2})$ as the hypothesis are not
satisfied in general. We will instead use Lemma 3.2 to show that for every $j$
there exists $i$ such that $T_{j}\nu=T_{i}\nu$ and also
$T_{j}^{2}\nu=T_{i}\nu$.

In view of the first condition in Lemma 3.2 it is convenient to introduce the
following notion:

######  Definition 3.3.

Let $D\subset\mathbb{G}^{\ast}$ and $\varepsilon>0$. A map
$\alpha:D\to\mathbb{G}^{\ast}$ is an _$\varepsilon$ -map_ if $\alpha(d)\in
dS_{\varepsilon}$ for every $d\in D$.

In the following lemma we find, for each $j\in[k]$, some $i\in[k]$ (which
might be equal to $j$) satisfying some technical property. In the sequel we
show that whenever $j$ and $i$ satisfy this technical property, then both
$T_{j}\nu=T_{i}\nu$ and $T_{j}^{2}\nu=T_{i}\nu$ for every empirical measure
$\nu$.

######  Lemma 3.4.

Let $(X,\tau)$ and $T_{1},\dots,T_{k}$ be as above. Then for every $j\in[k]$
there exists $i\in[k]$ satisfying the following property.

$\begin{split}&\text{For every $\varepsilon>0$ there exist subsets
$F,D\subset\mathbb{P}\cap\tau^{-1}(T_{j})$, where $F$ is finite with}\\\
&\text{$\log(F)>1/\varepsilon$ and $D$ is divergent, and an injective
$\varepsilon$-map $\alpha:FD\to\mathbb{P}\cap\tau^{-1}(T_{i})$.}\end{split}$
($\star$)

The next lemma shows that if $i$ and $j$ satisfy the property ($\star$ ‣ 3.4)
described in Lemma 3.4, then $T_{i}\nu=T_{j}\nu$ for every empirical measure
$\nu$.

######  Lemma 3.5.

Let $(X,\tau)$ and $T_{1},\dots,T_{k}$ be as above and let $\nu$ be an
empirical measure. Suppose $i,j\in[k]$ and for every $\varepsilon>0$ there
exist a divergent set $D\subset\mathbb{P}\cap\tau^{-1}(T_{j})$, some number
$m\in\mathbb{G}^{\ast}$ and an injective $\varepsilon$-map
$\alpha:mD\to\mathbb{P}\cap\tau^{-1}(T_{i})$. Then $T_{i}\nu=T_{j}\nu$.

Finally, the next lemma shows that whenever $i$ and $j$ satisfy the property
($\star$ ‣ 3.4) described in Lemma 3.4, then $T_{i}\nu=T_{j}^{2}\nu$ for every
empirical measure $\nu$.

######  Lemma 3.6.

Let $(X,\tau)$ and $T_{1},\dots,T_{k}$ be as above and let $\nu$ be an
empirical measure. If $i,j\in[k]$ satisfy ($\star$ ‣ 3.4), then
$T_{i}\nu=T_{j}^{2}\nu$.

The proofs of the various lemmas above are postponed to future subsections.
For now we verify that together they imply Theorem 3.1.

###### Proof of Theorem 3.1.

Given $j\in[k]$, use Lemma 3.4 to find $i\in[k]$ so that property ($\star$ ‣
3.4) holds. Then the conditions of Lemma 3.5 hold (taking an arbitrary $m\in
F$) and hence $T_{i}\nu=T_{j}\nu$. Finally, Lemma 3.6 implies that
$T_{i}\nu=T_{j}^{2}\nu$ and hence we conclude that
$T_{j}\nu=T_{i}\nu=T_{j}^{2}\nu$, finishing the proof. ∎

###  3.2. Proof of Lemma 3.2

First, we need a technical lemma, which makes use of the fact that the Følner
sequences we consider are dilated.

######  Lemma 3.7.

Let $\delta>0$ and let $\Phi=(\Phi_{N})_{N\in\mathbb{N}}$ be a dilated Følner
sequence. Then there exists $\varepsilon\in(0,\delta)$ such that for every
finite subset $F$ of $\mathbb{G}^{*}$, every $\varepsilon$-map
$\alpha:F\to\mathbb{G}^{\ast}$ and every function
$a:\mathbb{G}^{\ast}\to\mathbb{C}$ bounded by $1$, we have

$\limsup_{N\to\infty}\left|\mathop{\mathbb{E}}^{\log}_{p\in
F}\mathop{\mathbb{E}}_{n\in\Phi_{N}/p}a(n)-\mathop{\mathbb{E}}^{\log}_{q\in\alpha(F)}\mathop{\mathbb{E}}_{n\in\Phi_{N}/q}a(n)\right|\leq\delta.$

###### Proof.

Using Lemma 2.5, there exists $\varepsilon>0$ such that

$\limsup_{N\to\infty}\frac{\Big{|}\big{(}\Phi_{N}/p\big{)}\triangle\big{(}\Phi_{N}/q\big{)}\Big{|}}{\big{|}\Phi_{N}/p\big{|}}\leq\frac{\delta}{6}.$
(3.3)

whenever $q\in pS_{\varepsilon}$. If needed we can make $\varepsilon$ smaller
so that $|z-1|^{2}<\delta/3$ whenever $z\in S_{\varepsilon}$.

Now if $\alpha:F\to\mathbb{G}^{\ast}$ is an $\varepsilon$-map and
$a:\mathbb{G}^{\ast}\to\mathbb{D}$, then for every $p\in F$, by (3.3), we have
that

$\limsup_{N\to\infty}\left|\mathop{\mathbb{E}}_{n\in\Phi_{N}/p}a(n)-\mathop{\mathbb{E}}_{n\in\Phi_{N}/\alpha(p)}a(n)\right|\leq\frac{\delta}{3}.$
(3.4)

The conclusion now follows directly from Lemma A.3 with
$w(p)=1/\mathcal{N}(p)$, $v(p)=1/\mathcal{N}(\alpha(p))$,
$f(p)=\mathop{\mathbb{E}}_{n\in\Phi_{N}/p}a(n)$ and
$g(p)=\mathop{\mathbb{E}}_{n\in\Phi_{N}/\alpha(p)}a(n)$. ∎

The proof of Lemma 3.2 uses a version of the Turán-Kubiliys inequality for
Gaussian integers. The precise version we need and its proof are provided in
Appendix A for completeness.

###### Proof of Lemma 3.2.

Let $T,R\in{\mathcal{T}}$ be two commuting transformations satisfying the
hypothesis of the lemma. Let $\nu$ be an arbitrary additively empirical
measure and let $f\in C(X)$ with $\|f\|_{\infty}\leq 1$. We need to show that

$\int_{X}f\circ T\,\mathrm{d}\nu=\int_{X}f\circ R\,\mathrm{d}\nu.$

Since $\nu$ is an additively empirical measure, it suffices to show that for
every $x\in X$, every dilated Følner sequence $(\Phi_{N})_{N\in\mathbb{N}}$
and every $\delta>0$,

$\limsup_{N\to\infty}\left|\mathop{\mathbb{E}}_{n\in\Phi_{N}}f(\tau(n)Tx)-\mathop{\mathbb{E}}_{n\in\Phi_{N}}f(\tau(n)Rx)\right|<4\delta.$
(3.5)

Fix $\delta>0$ and let $\varepsilon\in(0,\delta)$ be given by Lemma 3.7. Let
$F\subset\tau^{-1}(T)$ and $\alpha\colon F\to\mathbb{G}^{\ast}$ satisfy the
lemma’s hypothesis.

Note that if $p\in\alpha(F)\subset\tau^{-1}(R)$, then for any
$n\in\mathbb{G}^{*}$, $\tau(np)=\tau(n)R$. Therefore, by applying the Turán-
Kublius inequality (Lemma A.1) with $a(n)=f(\tau(n)Tx)$ and $B=\alpha(F)$, we
have

$\limsup_{N\to\infty}\left|\mathop{\mathbb{E}}_{n\in\Phi_{N}}f(\tau(n)Tx)-\mathop{\mathbb{E}}_{p\in\alpha(F)}^{\log}\mathop{\mathbb{E}}_{n\in\Phi_{N}/p}f(\tau(n)TRx)\right|^{2}\leq\mathop{\mathbb{E}}_{p,q\in\alpha(F)}^{\log}\mathcal{N}(\gcd(p,q))-1\leq\varepsilon\leq\delta$
(3.6)

Similarly, by using Lemma A.1 with $a(n)=f(\tau(n)Rx)$ and $B=F$, we get

$\limsup_{N\to\infty}\left|\mathop{\mathbb{E}}_{n\in\Phi_{N}}f(\tau(n)Rx)-\mathop{\mathbb{E}}_{p\in
F}^{\log}\mathop{\mathbb{E}}_{n\in\Phi_{N}/p}f(\tau(n)TRx)\right|^{2}\leq\mathop{\mathbb{E}}_{p,q\in
F}^{\log}\mathcal{N}(\gcd(p,q))-1\leq\varepsilon\leq\delta.$ (3.7)

By Lemma 3.7, we have

$\limsup_{N\to\infty}\left|\mathop{\mathbb{E}}_{p\in\alpha(F)}^{\log}\mathop{\mathbb{E}}_{n\in\Phi_{N}/p}f(\tau(n)TRx)-\mathop{\mathbb{E}}_{p\in
F}^{\log}\mathop{\mathbb{E}}_{n\in\Phi_{N}/p}f(\tau(n)TRx)\right|\leq
2\delta.$ (3.8)

Relations (3.6), (3.7), and (3.8) give (3.5), finishing the proof. ∎

###  3.3. Proof of Lemma 3.6

We will use Lemma 3.2 directly with $T=T_{j}^{2}$ and $R=T_{i}$.

Let $\varepsilon>0$. Without loss of generality we can assume that
$\varepsilon<1$. Using ($\star$ ‣ 3.4), we can find sets
$F,D\subset\mathbb{P}\cap\tau^{-1}(T_{j})$ such that $F$ is finite with
$\log(F)>1/\varepsilon$, $D$ is divergent, and there is an injective
$\varepsilon$-map $\alpha:FD\to\mathbb{P}\cap\tau^{-1}(T_{i})$. Since $F$ is
finite and $D$ is divergent, we can find a finite set $F_{1}\subset D$ which
is disjoint from $F$ and satisfies $\log(F_{1})>1/\varepsilon$. Let
$F_{2}=FF_{1}$. The restriction of $\alpha$ to $F_{2}$ is an
$\varepsilon$-map, and by construction $\tau(F_{2})=\\{T_{j}^{2}\\}$ and
$\tau(\alpha(F_{2}))=\\{T_{i}\\}$.

One can compute that

$\log(\alpha(F_{2}))=\sum_{n\in
F_{2}}\frac{1}{\mathcal{N}(\alpha(n))}\geq\sum_{n\in
F_{2}}\frac{1}{4\mathcal{N}(n)}=\frac{\log(F_{2})}{4}=\frac{\log(F)\log(F_{1})}{4}\geq\frac{1}{4\varepsilon^{2}}.$

Finally, using Lemma A.2, we deduce that

$\mathop{\mathbb{E}}^{\log}_{n,m\in
F_{2}}\gcd(n,m)<(1+8\varepsilon)^{2}<1+80\varepsilon,\qquad\text{ and
}\qquad\mathop{\mathbb{E}}^{\log}_{n,m\in\alpha(F_{2})}\gcd(n,m)<1+16\varepsilon^{2}.$

Since $\varepsilon$ can be taken arbitrarily small, we meet the conditions of
Lemma 3.2, and hence conclude that $T_{j}^{2}\nu=T_{i}\nu$ as desired.

###  3.4. Sparse sets

By assumption, when restricted to primes, $\tau$ takes only finitely many
values. Moreover, $\tau(p)\in\\{T_{1},\dots,T_{k}\\}$ for all primes $p$
outside a convergent set. Throughout the proof we also need to deal with
numbers that are not primes, and we seek to find injective $\varepsilon$-maps
into sets of primes. For that it is important to know the value of $\tau$ on
the primes “near” a given $n$. For this purpose, it is convenient to introduce
the following colorings of $\mathbb{G}^{*}$: for each $\delta>0$ we define
$\chi=\chi_{\delta}:\mathbb{G}^{*}\to\\{0,1,\dots,k\\}$ by letting $\chi(n)$
be an arbitrary $\ell\in[k]$ satisfying

$\big{|}\mathbb{P}\cap
nS_{\delta}\cap\tau^{-1}(T_{\ell})\big{|}\geq\frac{3.5\delta^{2}\mathcal{N}(n)}{k\log\mathcal{N}(n)}.$

If no such $\ell\in[k]$ exists, we let $\chi(n)=0$. Note that, in view of the
prime number theorem (2.4), for each $\delta>0$ only finitely many
$n\in\mathbb{G}^{*}$ get colored $0$.

We want to build injective $\delta$-maps from certain monochromatic divergent
sets $D\subset\chi^{-1}(\ell)$ to the set $\mathbb{P}\cap\tau^{-1}(T_{\ell})$.
To get injectivity we need to make sure that the elements of $D$ are not too
clumped together; indeed some divergent sets have no divergent subset with an
injective $\delta$-map into the primes (see Example 3.10 below). This forces
us to introduce the notion of sparse sets:

######  Definition 3.8.

For $\delta>0$, we say an infinite set $D\subset\mathbb{G}$ is _$\delta$
-sparse_ if

$|D\cap
nS_{3\delta}|<\frac{\mathcal{N}(n)}{\log\mathcal{N}(n)}\left(\frac{3\delta^{2}}{k}+o_{\mathcal{N}(n)\to\infty}(1)\right).$
(3.9)

In the next lemma, sparcity is used to find $\delta$-maps into sets of primes.

######  Lemma 3.9.

Let $\delta>0$ and suppose $D\subset\mathbb{G}^{\ast}$ is an infinite
$\delta$-sparse set such that $\chi_{\delta}(D)=\\{j\\}$ for some $j\in[k]$.
Then there exists a co-finite subset $D^{\prime}\subset D$ and an injective
$\delta$-map $\alpha:D^{\prime}\to\mathbb{P}\cap\tau^{-1}(T_{j})$.

###### Proof.

Since $D$ is $\delta$-sparse, there exists a co-finite subset
$D^{\prime}\subset D$ such that

$|D^{\prime}\cap
nS_{3\delta}|<\frac{3.5\delta^{2}\mathcal{N}(n)}{k\log\mathcal{N}(n)}\text{
for every }n\in\mathbb{G}^{*}.$ (3.10)

For convenience denote $\chi:=\chi_{\delta}$ and enumerate
$D^{\prime}=\\{n_{1},n_{2},\dots\\}$. Since $\chi(n_{1})=j$, there is some
prime $p_{1}\in\mathbb{P}\cap\tau^{-1}(T_{j})$ such that $p_{1}\in
n_{1}S_{\delta}$. Let $\alpha(n_{1})=p_{1}$. Continuing in this manner, we
seek to define recursively, for each $i>1$, $\alpha(n_{i})$ to be any prime
$p_{i}\in\mathbb{P}\cap\tau^{-1}(T_{j})\setminus\\{\alpha(n_{1}),\dots,\alpha(n_{i-1})\\}$
with $p_{i}\in n_{i}S_{\delta}$. We claim that such $p_{i}$ exists. Since
$\chi(n_{i})=j$, there are at least
$\frac{3.5\delta^{2}\mathcal{N}(n_{i})}{k\log\mathcal{N}(n_{i})}$ many primes
$p$ in $\tau^{-1}(T_{j})\cap n_{i}S_{\delta}$. Let

$F_{i}:=\big{\\{}n_{t}:t<i\text{ and }\alpha(n_{t})\in
n_{i}S_{\delta}\big{\\}}.$

For each $n_{t}\in F_{i}$, we have $\alpha(n_{t})\in n_{t}S_{\delta}\cap
n_{i}S_{\delta}$, and so this intersection is non-empty. Thus,

$n_{t}\in\alpha(n_{t})S_{\delta}^{-1}\subset
n_{i}S_{\delta}S_{\delta}^{-1}\subset n_{i}S_{3\delta}.$

It follows that $F_{i}\subset D^{\prime}\cap n_{i}S_{3\delta}$. Since
$D^{\prime}$ satisfies (3.10), we have
$|F_{i}|<\frac{3.5\delta^{2}\mathcal{N}(n_{i})}{k\log\mathcal{N}(n_{i})}$.
Hence, there must be a prime $p$ other than
$\alpha(n_{1}),\dots,\alpha(n_{i-1})$ belonging to $\tau^{-1}(T_{j})\cap
n_{i}S_{\delta}$.

It is not hard to see that $\alpha$ is an injective $\,\mathrm{d}$-map on
$D^{\prime}$. ∎

Unfortunately, it is not true that every divergent set contains a sparse
divergent subset, as shown by the next example.

######  Example 3.10.

Fix $\delta>0$, $A\subset\mathbb{G}^{\ast}$, and let

$E_{A}:=\bigcup_{n\in A}\big{(}\mathbb{G}\cap nS_{\delta}\big{)}.$

It is not difficult to compute that $\log\big{(}\mathbb{G}\cap
nS_{\delta}\big{)}\approx\delta^{2}$. Therefore, if the sets $nS_{\delta}$
with $n\in A$ are pairwise disjoint then $E_{A}$ is divergent as long as $A$
is infinite. On the other hand, if $S$ is a $\delta$-sparse set, then

$\log(S\cap nS_{\delta})\ll_{\delta}\frac{1}{\mathcal{N}(n)}\big{|}S\cap
nS_{\delta}\big{|}\ll_{\delta}\frac{1}{k\log\mathcal{N}(n)},$

so if $\sum_{n\in A}\frac{1}{\log\mathcal{N}(n)}<\infty$, then any
$\delta$-sparse subset of $E_{A}$ must be convergent.

However, the next lemma shows that any divergent set of primes has a sparse
divergent subset. In fact, we need something stronger.

######  Lemma 3.11.

Let $D\subset\mathbb{P}$ be a divergent set, let $F\subset\mathbb{G}^{*}$ be a
finite set and let $\delta\in(0,1/3)$. Suppose that

$\forall n,m\in F,\qquad n\in mS_{10\delta}\quad\Rightarrow\quad n=m.$ (3.11)

Then there exists a divergent subset $D^{\prime}\subset D$ such that the
product set $FD^{\prime}$ satisfies

$\forall n\in\mathbb{G}^{\ast},\qquad|FD^{\prime}\cap
nS_{3\delta}|<\frac{3\delta^{2}\mathcal{N}(n)}{k\log\mathcal{N}(n)}.$ (3.12)

In particular, $FD^{\prime}$ is $\delta$-sparse.

###### Proof.

Consider the set $S_{1}:=(F\cup\\{1\\})S_{3\delta}$ and let
$S_{2}=S_{1}S_{1}^{-1}$ and $S=S_{2}S_{2}^{-1}$. Before we proceed with the
proof we make some observations that will be useful. Note that $1\in S_{1}$ so
$S_{1}\subset S_{2}\subset S$. The set $F$ is finite, so
$C_{0}:=\sup\\{|z|^{2}:z\in S\\}$ is a positive and finite number that depends
only on $\delta$ and $F$. The inverse $S^{-1}$ equals $S$, so for
$n,m\in\mathbb{G}$ the statements $n\in mS$ and $m\in nS$ are equivalent.
Since $S$ is bounded, it follows from (2.1) that

$\forall n\in\mathbb{G}^{\ast},\qquad\big{|}\mathbb{P}\cap nS\big{|}\leq
C_{1}\frac{\mathcal{N}(n)}{\log\mathcal{N}(n)}$

for some constant $C_{1}>0$ that depends only on $\delta$ and $F$. Moreover,
for any $m,n\in\mathbb{G}$ with $m\in nS$, we have
$\mathcal{N}(m)\geq\mathcal{N}(n)/C_{0}$. So

$\forall n\in\mathbb{G}^{\ast},\qquad\log\big{(}\mathbb{P}\cap
nS\big{)}=\sum_{p\in\mathbb{P}\cap
nS}\frac{1}{\mathcal{N}(p)}\leq\frac{\big{|}\mathbb{P}\cap
nS\big{|}}{\mathcal{N}(n)/C_{0}}\leq\frac{C_{0}C_{1}}{\log\mathcal{N}(n)}.$

Therefore, for any divergent set $D_{0}\subset\mathbb{P}$, any subset
$D_{1}\subset D_{0}$ satisfying $D_{0}\subset D_{1}S$ must satisfy $\sum_{d\in
D_{1}}\frac{1}{\log\mathcal{N}(d)}=\infty$.

We are now ready to proceed with the proof. Construct $D^{\prime}$ using a
greedy algorithm as follows: Enumerate the elements of
$D=\\{d_{1},d_{2},\dots\\}$ so that
$\mathcal{N}(d_{i})\leq\mathcal{N}(d_{i+1})$. Let $D_{0}^{\prime}=\emptyset$
and for each $i\in\mathbb{N}$ let
$D_{i}^{\prime}:=D_{i-1}^{\prime}\cup\\{d_{i}\\}$ if it satisfies (3.12), or
$D_{i}^{\prime}:=D_{i-1}^{\prime}$ otherwise. Take $D^{\prime}=\bigcup
D_{i}^{\prime}$. Note that $D^{\prime}$ satisfies (3.12), but for any $d\in
D\setminus D^{\prime}$, the set $D^{\prime}\cup\\{d\\}$ does not. We will show
that the set $D^{\prime}$ thus constructed is divergent, and this will finish
the proof.

Let $D_{0}=D\setminus D^{\prime}$. If $D_{0}$ is not divergent then
$D^{\prime}$ must be divergent, finishing the proof, so we will now assume
that $D_{0}$ is a divergent subset of $\mathbb{P}$. Using again a greedy
algorithm, we may find a maximal subset $D_{1}\subset D_{0}$ satisfying

$\forall n,m\in D_{1},\qquad n\in mS\quad\Rightarrow\quad n=m.$ (3.13)

Since $D_{1}$ is maximal we have $D_{0}\subset D_{1}S$, so it follows from the
observations at the beginning of the proof that $\sum_{d\in
D_{1}}\frac{1}{\log\mathcal{N}(d)}=\infty$.

We claim that for each $d\in D_{1}$, $\log(D^{\prime}\cap
dS_{2})\geq\frac{C_{2}}{\log\mathcal{N}(d)}$, where $C_{2}>0$ is a constant
that only depends on $\delta$, $F$ and $k$. Note that (3.13) and the fact that
$S=S_{2}S_{2}^{-1}$ imply that the sets $dS_{2}$ with $d\in D_{1}$ are
pairwise disjoint. Therefore if we prove this claim, then we finish the proof
that $\log(D^{\prime})=\infty$.

Since $d\in D_{1}\subset D\setminus D^{\prime}$ and $D^{\prime}$ is a maximal
subset of $D$ satisfying (3.12), it follows that
$\tilde{D}:=D^{\prime}\cup\\{d\\}$ does not satisfy (3.12) and hence there
exists $n\in\mathbb{G}^{*}$ such that the set $A:=F\tilde{D}\cap nS_{3\delta}$
satisfies

$|A|\geq\frac{3\delta^{2}\mathcal{N}(n)}{k\log\mathcal{N}(n)}.$

Observe that necessarily $n\in FdS_{3\delta}^{-1}$; indeed since $D^{\prime}$
does satisfy (3.12), there exists $a_{0}\in A\cap Fd$. Since $A\subset
nS_{3\delta}$, it follows that $n\in a_{0}S_{3\delta}^{-1}\subset
FdS_{3\delta}^{-1}$.

For each $a\in A$, there exists $\tilde{a}\in\tilde{D}$ such that $a\in
F\tilde{a}$. In view of (3.11) (and the fact that $A\subset nS_{3\delta}$),
the map $a\mapsto\tilde{a}$ is injective, so $a_{0}$ is the only element of
$A$ with $\tilde{a_{0}}=d$. Letting $\tilde{A}:=\big{\\{}\tilde{a}:a\in
A\big{\\}}\setminus\\{d\\}$ we have $|\tilde{A}|=|A|-1$ and $\tilde{A}\subset
D^{\prime}$. Moreover,

$\tilde{A}\subset AF^{-1}\subset nS_{3\delta}F^{-1}\subset
FdS_{3\delta}^{-1}S_{3\delta}F^{-1}\subset dS_{2},$

so $\tilde{A}\subset D^{\prime}\cap dS_{2}$ and we’ve reduced the claim to
showing that $\log(\tilde{A})\geq\frac{C_{2}}{\log\mathcal{N}(d)}$. By
removing from $D$ a finite set (depending only on $\delta,k$ and $F$), we may
assume that $\mathcal{N}(n)$ is large enough so that $|A|>2$, and hence
$|\tilde{A}|\geq|A|/2$. Also, since $\tilde{A}\subset
nS_{3\delta}F^{-1}\subset nS$, it follows that for every
$\tilde{a}\in\tilde{A}$, we have the estimate
$\mathcal{N}(\tilde{a})\leq\mathcal{N}(n)C_{0}$. Similarly, from $n\in
FdS_{3\delta}^{-1}\subset dS$ it follows that
$\mathcal{N}(n)\leq\mathcal{N}(d)C_{0}$. By further removing a finite set
(depending only on $\delta,k$ and $F$) from $D$, we may assume that
$\mathcal{N}(n)\leq\mathcal{N}(d)^{2}$. We conclude that

$\log(\tilde{A})=\sum_{\tilde{a}\in\tilde{A}}\frac{1}{\mathcal{N}(\tilde{a})}\geq\frac{|\tilde{A}|}{\max\\{\mathcal{N}(\tilde{a}):\tilde{a}\in\tilde{A}\\}}\geq\frac{|A|}{2C_{0}\mathcal{N}(n)}\geq\frac{3\delta^{2}}{2C_{0}k\log\mathcal{N}(n)}\geq\frac{3\delta^{2}}{4C_{0}k\log\mathcal{N}(d)}.$
(3.14)

This proves the claim with $C_{2}=\frac{3\delta^{2}}{4C_{0}k}$. ∎

Applying Lemma 3.11 for the set $F=\\{1\\}$, we deduced the following:

######  Corollary 3.12.

For every $\delta>0$, any divergent set of $\mathbb{P}$ has a $\delta$-sparse
divergent subset.

###  3.5. Proof of Lemma 3.4

Let $(X,\tau),T_{1},\dots,T_{k}$ and $j\in[k]$ be as in the statement of the
lemma. Let $I\subset[k]$ be the set of those $i\in[k]$ for which ($\star$ ‣
3.4) does not hold. We want to show that $I$ is not all of $[k]$. For each
$\ell\in I$ there is some $\varepsilon_{\ell}$ for which ($\star$ ‣ 3.4)
fails. Take $\varepsilon>0$ smaller than all the $\varepsilon_{\ell}$. We will
find some $i\in[k]$ for which ($\star$ ‣ 3.4) holds with this choice of
$\varepsilon$. By construction, such $i$ will not be in $I$, finishing the
proof.

With the choice of $\varepsilon$ from the previous paragraph, take a finite
set $\tilde{F}\subset\mathbb{P}\cap\tau^{-1}(T_{j})$ with
$\log(\tilde{F})>k/\varepsilon$. Pick $\delta>0$ sufficiently small so that
the ratio between any two distinct elements of $\tilde{F}$ is outside
$S_{10\delta}$. We may also require that $\delta<\min\\{1/3,\varepsilon\\}$.
Consider the coloring $\chi=\chi_{\delta}:\mathbb{G}^{*}\to\\{0,\dots,k\\}$
described in Section 3.4.

Recall that $\chi(n)=0$ for only finitely many $n\in\mathbb{G}^{*}$.
Therefore, for each sufficiently large $n\in\mathbb{P}\cap\tau^{-1}(T_{j})$,
we may find a subset $F_{n}\subset\tilde{F}$ with $\log(F_{n})>1/\varepsilon$
and such that $nF_{n}$ is monochromatic with a color in $[k]$. Since there are
only finitely many possible subsets of $\tilde{F}$, we can find a subset
$F\subset\tilde{F}$ with $\log(F)>1/\varepsilon$ and a divergent set
$D_{1}\subset\mathbb{P}\cap\tau^{-1}(T_{j})$ such that $F_{n}=F$ for every
$n\in D_{1}$. We may then find a color $i\in[k]$ and a divergent subset
$D_{2}\subset D_{1}$ such that $\chi(Fd)=\\{i\\}$ for all $d\in D_{2}$. In
other words, $\chi(FD_{2})=\\{i\\}$ for this choice of $D_{2}$.

We can now apply Lemma 3.11 to find a subset $D\subset D_{2}$ which is
divergent and such that $FD$ is $\delta$-sparse. Finally, we apply Lemma 3.9
to find a $\delta$-map $\alpha:FD^{\prime}\to\mathbb{P}\cap\tau^{-1}(T_{i})$
for some co-finite subset $D^{\prime}$ of $D$. We conclude that ($\star$ ‣
3.4) holds with the choice of $\varepsilon$ in the first paragraph, which
implies that $i\notin I$ and hence finishes the proof.

###  3.6. Proof of Lemma 3.5

In this section, we fix a function
$\lfloor\cdot\rfloor:\mathbb{C}\to\mathbb{G}$ satisfying $\big{|}\lfloor
z\rfloor-z\big{|}<1$ for all $z\in\mathbb{C}$. Given a set
$S\subset\mathbb{C}$, we denote by $\lfloor S\rfloor$ the set $\\{\lfloor
s\rfloor:s\in S\\}$.

######  Lemma 3.13.

Let $\delta>0$ and let $D\subset\mathbb{G}$ be a $\delta$-sparse set. If
$z\in\mathbb{C}$ with $|z|\geq 1$, then $\lfloor zD\rfloor$ is also
$\delta$-sparse.

###### Proof.

For each $n\in\mathbb{G}^{\ast}$, note that

$\big{\\{}u\in\mathbb{C}:\lfloor u\rfloor\in nS_{3\delta}\big{\\}}\subset
nS_{3\delta}+\mathbb{D},$

and hence for all $d\in D$ with $\lfloor zd\rfloor\in nS_{3\delta}$, we have
that

$d\in\frac{n}{z}S_{3\delta}+\frac{1}{|z|}\mathbb{D}.$

Since the number of lattice points in

$\left(\frac{n}{z}S_{3\delta}+\frac{1}{|z|}\mathbb{D}\right)\setminus\left\lfloor\frac{n}{z}\right\rfloor
S_{3\delta}$

can be bounded from above by $C\sqrt{\mathcal{N}(n)}$ for some constant $C$
that only depends on $\delta$, we conclude that

$\displaystyle\big{|}\lfloor zD\rfloor\cap nS_{3\delta}\big{|}$
$\displaystyle\leq$
$\displaystyle\left|D\cap\left\lfloor\frac{n}{z}\right\rfloor
S_{3\delta}\right|+C\sqrt{\mathcal{N}(n)}$ $\displaystyle\leq$
$\displaystyle\frac{\mathcal{N}\left(\left\lfloor
n/z\right\rfloor\right)}{\log\mathcal{N}\left(\left\lfloor
n/z\right\rfloor\right)}\left(\frac{3\delta^{2}}{k}+o_{\mathcal{N}(n)\to\infty}(1)\right)$
$\displaystyle\leq$
$\displaystyle\frac{\mathcal{N}(n)}{\log\mathcal{N}(n)}\left(\frac{3\delta^{2}}{k}+o_{\mathcal{N}(n)\to\infty}(1)\right).$

∎

###### Proof of Lemma 3.5.

Let $i,j\in[k]$ satisfy the hypothesis of Lemma 3.5. Fix an empirical measure
$\nu$ and denote by $K=\\{\ell\in[k]:T_{\ell}\nu=T_{j}\nu\\}$; we want to show
that $i\in K$. For each pair $(\ell,\ell^{\prime})\in[k]^{2}$ with $\ell\in K$
and $\ell^{\prime}\notin K$ we have $T_{\ell}\nu=T_{j}\nu\neq
T_{\ell^{\prime}}\nu$. Therefore, in view of Lemma 3.2, for each such pair,
there exists some $\varepsilon_{\ell,\ell^{\prime}}>0$ for which the
conditions in Lemma 3.2 do not hold. Let

$\varepsilon=\min\\{\varepsilon_{\ell,\ell^{\prime}}:\ell\in
K,\ell^{\prime}\in[k]\setminus K\\}.$

Using the fact that there are only finitely many such pairs, we deduce that
$\varepsilon>0$. Moreover, in view of Lemma A.2, $\varepsilon$ satisfies the
following property:

Whenever $\ell\in K$ and $\ell^{\prime}\in[k]$, if there is an injective
$\varepsilon$-map between a divergent subset of
$\mathbb{P}\cap\tau^{-1}(R_{\ell})$ and
$\mathbb{P}\cap\tau^{-1}(R_{\ell^{\prime}})$, then also $\ell^{\prime}\in K$.
(P)

Next let $\delta>0$ be much smaller than $\varepsilon$ (the exact magnitude of
$\delta$ will be determined later as a function of $\varepsilon$ and $k$). By
the assumption of the lemma, there exists a divergent set
$D\subset\mathbb{P}\cap\tau^{-1}(T_{j})$, some $m\in\mathbb{G}^{\ast}$ and an
injective $\delta$-map $\alpha:mD\to\mathbb{P}\cap\tau^{-1}(T_{i})$. Let $z\in
S_{\delta}$ and $r\in\mathbb{N}$ be such that $z^{r}=m$. Note that $|z|\geq
1$. Also let $\chi=\chi_{\delta}$ be the coloring introduced in Section 3.4.

In view of Corollary 3.12, we can replace $D$ with a divergent subset which is
$\delta$-sparse. Lemma 3.13 then implies that $\lfloor z^{t}D\rfloor$ is
$\delta$-sparse for all $t\in\\{0,1,\ldots,r\\}$. We want that, for each
$t\in\\{0,\dots,r\\}$, the map $\beta_{t}\colon\lfloor z^{t}D\rfloor\to\lfloor
z^{t+1}D\rfloor$, $\lfloor z^{t}n\rfloor\mapsto\lfloor z^{t+1}n\rfloor$ is a
well defined, injective map. Since $\big{|}\lfloor z_{1}\rfloor-\lfloor
z_{2}\rfloor\big{|}>|z_{1}-z_{2}|-2$ for any $z_{1},z_{2}\in\mathbb{C}$, by
replacing $D$ with a divergent subset satisfying $|d_{1}-d_{2}|>2r+2$ for
every pair $d_{1},d_{2}\in D$, we make sure that indeed each of the maps
$\beta_{t}$ is injective and well defined. Since $r\in S_{\delta}$, after
removing from $D$ a finite set if necessary, each of the maps $\beta_{t}$ is a
$(2\delta)$-map.

Using the fact that being divergent is a partition regular property, we can
once again replace $D$ with a divergent subset so that the set $\lfloor
z^{t}D\rfloor$ is $\chi$-monochromatic for each $t\in\\{0,1,\ldots,r\\}$. Let
$\ell_{t}\in[k]$ be the (single) color of $\lfloor z^{t}D\rfloor$ for each
$t\in\\{1,\dots r\\}$. For convenience of notation, let $\ell_{0}:=j$, so that
$\ell_{0}\in K$ trivially. We will prove by induction that $\ell_{t}\in K$ for
all $t\in\\{0,\dots,r\\}$.

By Lemma 3.9, after passing to a co-finite subset of $D$ if necessary, for
each $t\in\\{0,\dots,r\\}$ there exists an injective $\delta$-map
$\tilde{\alpha}_{t}:\lfloor
z^{t}D\rfloor\to\mathbb{P}\cap\tau^{-1}(T_{\ell_{t}})$ (for $t=0$ we can take
$\tilde{\alpha}_{0}$ to be the identity map). Making $\delta$ small enough in
terms of $\varepsilon$, it follows that
$\tilde{\alpha}_{t+1}\circ\beta_{t}\circ\tilde{\alpha}_{t}^{-1}$ is an
injective $\varepsilon$-map between a divergent subset of
$\mathbb{P}\cap\tau^{-1}(T_{\ell_{t}})$ and
$\mathbb{P}\cap\tau^{-1}(T_{\ell_{t+1}})$ (where we only define
$\tilde{\alpha}_{t}^{-1}$ on those numbers which has a (unique) pre-image). In
view of (P), it follows by induction that $\ell_{t}\in K$ for all
$t\in\\{0,\dots,r\\}$.

In particular $\ell_{r}\in K$. Since $z^{r}D=mD$ and
$\alpha:mD\to\mathbb{P}\cap\tau^{-1}(T_{i})$ is an injective $\delta$-map, it
follows that $\alpha\circ\tilde{\alpha}_{r}^{-1}$ is an injective
$\varepsilon$-map between a divergent subset of
$\mathbb{P}\cap\tau^{-1}(T_{\ell_{r}})$ and $\mathbb{P}\cap\tau^{-1}(T_{i})$
(again we only define $\tilde{\alpha}_{r}^{-1}$ on those numbers which has a
(unique) pre-image). Using (P) one final time, we conclude that $i\in K$ as
desired. ∎

##  4\. Applications

In this section we deduce from E other results formulated in the introduction.
For a function $f\colon\mathbb{G}^{\ast}\to\mathbb{C}$, we say that
$\mathop{\mathbb{E}}(f)$ exists if $\mathop{\mathbb{E}}_{n\in\Phi_{N}}f(n)$
converges to the same limit for every dilated Følner sequence
$(\Phi_{N})_{N\in\mathbb{N}}$. In this case, we use $\mathop{\mathbb{E}}(f)$
to denote the common limit.

###  4.1. Comparing the averages of two multiplicative functions

In this section we present some preliminary estimates needed for the proof of
C.

######  Lemma 4.1.

For any dilated Følner sequence $(\Phi_{N})_{N\in\mathbb{N}}$, there exist
constants $C,N_{0}>0$ such that if $f,g:\mathbb{G}^{\ast}\to\mathbb{C}$ are
completely multiplicative functions bounded by $1$, then for all $N\geq
N_{0}$,

$\left|\mathop{\mathbb{E}}_{n\in\Phi_{N}}f(n)-g(n)\right|<C\sum_{p\in\mathbb{P}}\frac{|f(p)-g(p)|}{\mathcal{N}(p)}.$

###### Proof.

Since $(\Phi_{N})_{N\in\mathbb{N}}$ is a dilated Følner sequence we have
$\Phi_{N}=\mathbb{G}^{\ast}\cap k_{N}U$ where $U$ is a Jordan measurable
subset of $\mathbb{C}$ and $(k_{N})_{N\in\mathbb{N}}$ is a sequence of
positive real numbers satisfying $k_{N}\to\infty$ as $N\to\infty$. Let $R>0$
be such that the disk $R\mathbb{D}$ contains $U$. Note that
$\Phi_{N}\subset\mathbb{G}^{*}\cap k_{N}R\mathbb{D}$ for all $N$ and

$\lim_{N\to\infty}\frac{|\Phi_{N}|}{|\mathbb{G}^{*}\cap
k_{N}R\mathbb{D}|}=\frac{\operatorname{m}(U)}{\operatorname{m}(R\mathbb{D})}>0,$

where $\operatorname{m}$ denotes the Lebesgue measure. Thus there are positive
constants $\rho,N_{0}$ such that $|\Phi_{N}|>\rho|\mathbb{G}^{*}\cap
k_{N}R\mathbb{D}|$ for all $N\geq N_{0}$. It follows that for any
$a\in\mathbb{G}^{\ast}$,

$\left|\Phi_{N}/a\right|\leq\left|(\mathbb{G}^{\ast}\cap
k_{N}R\mathbb{D})/a\right|\leq\frac{2|\mathbb{G}^{\ast}\cap
k_{N}R\mathbb{D}|}{\mathcal{N}(a)}\leq\frac{2|\Phi_{N}|}{\rho\mathcal{N}(a)}.$
(4.1)

Let $f,g:\mathbb{G}^{\ast}\to\mathbb{C}$ be completely multiplicative
functions bounded by $1$. First, assume $f(q)=g(q)$ for all primes
$q\in\mathbb{P}$ except for $q=p$. We will show that for all $N\geq N_{0}$,

$\left|\mathop{\mathbb{E}}_{n\in\Phi_{N}}f(n)-g(n)\right|<\frac{C|f(p)-g(p)|}{\mathcal{N}(p)}.$
(4.2)

Indeed, since $f(n)=g(n)$ if $p\nmid n$, we have

$\left|\sum_{n\in\Phi_{N}}\big{(}f(n)-g(n)\big{)}\right|=\left|\sum_{k=0}^{\infty}\sum_{\begin{subarray}{c}n\in\Phi_{N}\\\
p^{k}\|n\\\
\end{subarray}}\big{(}f(n)-g(n)\big{)}\right|=\left|\sum_{k=0}^{\infty}\sum_{\begin{subarray}{c}n\in\Phi_{N}/p^{k}\\\
p\nmid n\end{subarray}}f(np^{k})-g(np^{k})\right|\\\
=\left|\sum_{k=0}^{\infty}\big{(}f(p^{k})-g(p^{k})\big{)}\sum_{\begin{subarray}{c}n\in\Phi_{N}/p^{k}\\\
p\nmid
n\end{subarray}}f(n)\right|\leq\sum_{k=1}^{\infty}\big{|}f(p^{k})-g(p^{k})\big{|}\cdot\big{|}\Phi_{N}/p^{k}\big{|}$

Using (4.1) and dividing by $|\Phi_{N}|$, we get that

$\left|\mathop{\mathbb{E}}_{n\in\Phi_{N}}\big{(}f(n)-g(n)\big{)}\right|<\sum_{k=1}^{\infty}\frac{|f(p)^{k}-g(p)^{k}|}{\rho\mathcal{N}(p)^{k}}=\frac{|f(p)-g(p)|}{\rho\mathcal{N}(p)}\sum_{k=1}^{\infty}\frac{|\sum_{j=0}^{k-1}f(p)^{j}g(p)^{k-1-j}|}{\mathcal{N}(p)^{k-1}}\\\
\leq\frac{|f(p)-g(p)|}{\rho\mathcal{N}(p)}\sum_{k=1}^{\infty}\frac{k}{\mathcal{N}(p)^{k-1}}\leq\frac{|f(p)-g(p)|}{\rho\mathcal{N}(p)}\sum_{k=1}^{\infty}\frac{k}{2^{k-1}}\leq\frac{4|f(p)-g(p)|}{\rho\mathcal{N}(p)},$

establishing (4.2) with $C=4/\rho$.

For arbitrary functions $f$ and $g$, we can apply (4.2) by changing $f$ at one
prime at a time. More precisely, fix $N>N_{0}$ and let
$\\{p_{1},\ldots,p_{s}\\}\subset\mathbb{P}$ be a list of pairwise co-primes
containing (up to a unit) every prime divisor of an element of $\Phi_{N}$.
Define completely multiplicative functions $f_{j}:\mathbb{G}^{*}\to\mathbb{C}$
inductively as follows: $f_{0}=f$ and for each $1\leq j\leq s$, let

$f_{j}(p)=\begin{cases}f_{j-1}(p)\text{ for
}p\in\mathbb{P}\setminus\\{p_{j}\\}\\\ g(p)\text{ for }p=p_{j}.\end{cases}$

Since $f_{j}(p)=f_{j-1}(p)$ for all primes $p$ except for $p=p_{j}$, using
(4.2) we deduce that

$\left|\mathop{\mathbb{E}}_{n\in\Phi_{N}}(f_{j}(n)-f_{j-1}(n))\right|<\frac{C|f(p_{j})-g(p_{j})|}{\mathcal{N}(p_{j})}.$

By construction, $f_{k}(n)=g(n)$ for all $n\in\Phi_{N}$. Thus, by the triangle
inequality,

$\left|\mathop{\mathbb{E}}_{n\in\Phi_{N}}(f(n)-g(n))\right|\leq\sum_{j=1}^{s}\left|\mathop{\mathbb{E}}_{n\in\Phi_{N}}(f_{j-1}(n)-f_{j}(n))\right|<C\sum_{j=1}^{s}\frac{|f(p_{j})-g(p_{j})|}{\mathcal{N}(p_{j})}<C\sum_{p\in\mathbb{P}}\frac{|f(p)-g(p)|}{\mathcal{N}(p)}.$

∎

######  Lemma 4.2.

Let $f,g:\mathbb{G}^{\ast}\to\mathbb{C}$ be completely multiplicative
functions bounded by $1$. Suppose $\mathop{\mathbb{E}}(g)$ exists and
$f(q)=g(q)$ for all but one prime $q=p$ and $f(\mathrm{i})=g(\mathrm{i})$.
Then $\mathop{\mathbb{E}}(f)$ exists and equals

$\frac{1-g(p)/\mathcal{N}(p)}{1-f(p)/\mathcal{N}(p)}\cdot\mathop{\mathbb{E}}(g).$

###### Proof.

Fix an arbitrary dilated Følner sequence $(\Phi_{N})_{N\in\mathbb{N}}$.

$\displaystyle\sum_{n\in\Phi_{N}}f(n)$ $\displaystyle=$
$\displaystyle\sum_{n\in\Phi_{N}}f(n)\sum_{k=0}^{\infty}1_{p^{k}||n}=\sum_{k=0}^{\infty}f(p)^{k}\left(\sum_{n\in\Phi_{N}/p^{k}}g(n)1_{p\nmid
n}\right)$ $\displaystyle=$
$\displaystyle\sum_{k=0}^{\infty}f(p)^{k}\left(\sum_{n\in\Phi_{N}/p^{k}}g(n)-g(p)\sum_{n\in\Phi_{N}/p^{k+1}}g(n)\right)$
$\displaystyle=$
$\displaystyle\sum_{n\in\Phi_{N}}g(n)+\big{(}f(p)-g(p)\big{)}\sum_{k=1}^{\infty}f(p)^{k-1}\sum_{n\in\Phi_{N}/p^{k}}g(n).$

Since $|\Phi_{N}/p^{k}|/|\Phi_{N}|\to\frac{1}{\mathcal{N}(p)^{k}}$ as
$N\to\infty$, it follows that, after dividing by $|\Phi_{N}|$ and taking the
limit as $N\to\infty$ above (and making use of the dominated convergence
theorem),

$\lim_{N\to\infty}\mathop{\mathbb{E}}_{n\in\Phi_{N}}f(n)=\mathop{\mathbb{E}}(g)+\frac{f(p)-g(p)}{\mathcal{N}(p)}\cdot\sum_{k=1}^{\infty}\frac{f(p)^{k-1}}{\mathcal{N}(p)^{k-1}}\mathop{\mathbb{E}}(g)=\frac{\mathcal{N}(p)-g(p)}{\mathcal{N}(p)-f(p)}\mathop{\mathbb{E}}(g).$

In particular, $\lim_{N\to\infty}\mathop{\mathbb{E}}_{n\in\Phi_{N}}f(n)$
exists and its value is independent of the Følner sequence
$(\Phi_{N})_{N\in\mathbb{N}}$, so $\mathop{\mathbb{E}}(f)$ exists. ∎

######  Lemma 4.3.

Let $f,g:\mathbb{G}^{\ast}\to\mathbb{C}$ be completely multiplicative
functions bounded by $1$ satisfying $f(\mathrm{i})=g(\mathrm{i})$. Suppose
$\mathop{\mathbb{E}}(g)$ exists and

$\sum_{p\in\mathbb{P}_{1}}\frac{|g(p)-f(p)|}{\mathcal{N}(p)}<\infty.$ (4.3)

Then $\mathop{\mathbb{E}}(f)$ exists and equals

$\mathop{\mathbb{E}}(g)\cdot\prod_{p\in\mathbb{P}_{1}}\frac{1-g(p)/\mathcal{N}(p)}{1-f(p)/\mathcal{N}(p)}.$

###### Proof.

First note that

$\prod_{p\in\mathbb{P}_{1}}\frac{1-g(p)/\mathcal{N}(p)}{1-f(p)/\mathcal{N}(p)}=\prod_{p\in\mathbb{P}_{1}}\left(1+\frac{f(p)-g(p)}{\mathcal{N}(p)-f(p)}\right)$
(4.4)

and hence converges in view of (4.3) and the fact that $|f(p)|\leq
1\leq\mathcal{N}(p)/2$ for every $p\in\mathbb{P}_{1}$.

Fix $\varepsilon>0$ and find a finite set $F\subset\mathbb{P}_{1}$ such that

$\left|\prod_{p\in
F}\frac{1-g(p)/\mathcal{N}(p)}{1-f(p)/\mathcal{N}(p)}-\prod_{p\in\mathbb{P}_{1}}\frac{1-g(p)/\mathcal{N}(p)}{1-f(p)/\mathcal{N}(p)}\right|<\varepsilon\qquad\text{and}\qquad\sum_{p\in\mathbb{P}_{1}\setminus
F}\frac{|g(p)-f(p)|}{\mathcal{N}(p)}<\varepsilon.$

Let $h:\mathbb{G}^{\ast}\to\mathbb{C}$ be the completely multiplicative
function satisfying $h(p)=f(p)$ for $p\in F$, $h(p)=g(p)$ for
$p\in\mathbb{P}_{1}\setminus F$, and $h(\mathrm{i})=1$. Applying Lemma 4.2
inductively $|F|$ times, it follows that $\mathop{\mathbb{E}}(h)$ exists and
equals

$\mathop{\mathbb{E}}(g)\cdot\prod_{p\in
F}\frac{1-g(p)/\mathcal{N}(p)}{1-f(p)/\mathcal{N}(p)}.$

On the other hand, in view of Lemma 4.1, for every dilated Følner sequence
$(\Phi_{N})_{N\in\mathbb{N}}$, there is a constant $C>0$ such that
$\big{|}\mathop{\mathbb{E}}_{n\in\Phi_{N}}(h(n)-f(n))\big{|}\leq C\varepsilon$
for a sufficiently large $N$. It follows that

$\limsup_{N\to\infty}\left|\mathop{\mathbb{E}}_{n\in\Phi_{N}}f(n)-\mathop{\mathbb{E}}(g)\cdot\prod_{p\in\mathbb{P}_{1}}\frac{1-g(p)/\mathcal{N}(p)}{1-f(p)/\mathcal{N}(p)}\right|<\big{(}C+|\mathop{\mathbb{E}}(g)|\big{)}\varepsilon.$

The conclusion now follows by taking $\varepsilon\to 0$. ∎

######  Corollary 4.4.

Let $g:\mathbb{G}^{\ast}\to[0,1]$ be a completely multiplicative function such
that $\sum_{p\in\mathbb{P}_{1}}\frac{|1-g(p)|}{\mathcal{N}(p)}=\infty$. Then
$\mathop{\mathbb{E}}(g)=0$.

###### Proof.

For each $M\in\mathbb{N}$, let $g_{M}:\mathbb{G}^{\ast}\to[0,1]$ be the
completely multiplicative function satisfying
$g_{M}(\mathrm{i})=1,g_{M}(p)=g(p)$ whenever $\mathcal{N}(p)\leq M$ and
$g_{M}(p)=1$ otherwise. Using Lemma 4.3 to compare $g_{M}$ with the constant
$1$ function we conclude that $\mathop{\mathbb{E}}(g_{M})$ exists and
satisfies

$\mathop{\mathbb{E}}(g_{M})=\prod_{\begin{subarray}{c}p\in\mathbb{P}_{1}\\\
\mathcal{N}(p)\leq
M\end{subarray}}\frac{1-1/\mathcal{N}(p)}{1-g(p)/\mathcal{N}(p)}\leq\prod_{\begin{subarray}{c}p\in\mathbb{P}_{1}\\\
\mathcal{N}(p)\leq
M\end{subarray}}\left(1-\frac{1-g(p)}{\mathcal{N}(p)}\right).$

Since $\sum_{p\in\mathbb{P}_{1}}\frac{|1-g(p)|}{\mathcal{N}(p)}=\infty$ it
follows that $\lim_{M\to\infty}\mathop{\mathbb{E}}(g_{M})=0$. On the other
hand, for all $M$ we have $g_{M}(n)\geq g(n)\geq 0$, so we conclude that
$\mathop{\mathbb{E}}(g)$ exists and equals $0$. ∎

###  4.2. Proof of C

Let $f:\mathbb{G}^{*}\to\mathbb{C}$ be a bounded completely multiplicative
function. Recall from (1.6) the infinite product $P(f)$:

$P(f)\coloneqq\prod_{p\in\mathbb{P}_{1}}\frac{\mathcal{N}(p)-1}{\mathcal{N}(p)-f(p)}.$

We first show that, if $\operatorname{Arg}(f(\mathbb{P}))$ is finite, then
$P(f)$ exists.

######  Lemma 4.5.

Let $f:\mathbb{G}^{\ast}\to\mathbb{C}$ be a completely multiplicative function
bounded by $1$ and suppose $\operatorname{Arg}(f(\mathbb{P}_{1}))$ is finite.
Then $P(f)$ exists and has norm at most 1; moreover, $P(f)=0$ if and only if

$\sum_{p\in\mathbb{P}_{1}}\frac{|1-f(p)|}{\mathcal{N}(p)}=\infty.$

###### Proof.

We will use the easy to check fact that $\prod(1+a_{i})$ converges for any
bounded complex valued sequence $(a_{i})$ for which $\sum|a_{i}|$ is finite.
Since

$\frac{\mathcal{N}(p)-1}{\mathcal{N}(p)-f(p)}=1-\frac{1-f(p)}{\mathcal{N}(p)-f(p)},\qquad\text{
and }\qquad\left|\frac{1-f(p)}{\mathcal{N}(p)-f(p)}\right|\leq
2\left|\frac{1-f(p)}{\mathcal{N}(p)}\right|$

it follows that $P(f)$ exists whenever
$\displaystyle\sum_{p\in\mathbb{P}_{1}}\frac{|1-f(p)|}{\mathcal{N}(p)}<\infty$.

On the other hand, if
$\displaystyle\sum_{p\in\mathbb{P}_{1}}\frac{|1-f(p)|}{\mathcal{N}(p)}=\infty$
and $\operatorname{Arg}(f(\mathbb{P}))$ is finite, there must exist $r<1$ such
that $Q:=\\{p\in\mathbb{P}_{1}:\mathrm{Re}\,f(p)<r\\}$ is divergent. We will
show that, in this case, $P(f)=0$ (and in particular it exists). Since
$\left|\frac{\mathcal{N}(p)-1}{\mathcal{N}(p)-f(p)}\right|\leq 1$ for all $p$,
if suffices to show that $\prod_{p\in
Q}\frac{\mathcal{N}(p)-1}{\mathcal{N}(p)-f(p)}=0$.

Note that

$\frac{\mathcal{N}(p)-1}{\mathcal{N}(p)-f(p)}=1-\frac{1-f(p)}{\mathcal{N}(p)-f(p)}\qquad\text{
and
}\qquad\mathrm{Re}\,\frac{1-f(p)}{\mathcal{N}(p)-f(p)}\geq\mathrm{Re}\,\frac{1-f(p)}{2\mathcal{N}(p)}\geq\frac{1-r}{2\mathcal{N}(p)}$

for every $p\in Q$. Because $Q$ is divergent, the desired conclusion is now
true thanks to the fact that $\prod(1-a_{i})=0$ whenever $(a_{i})$ is a
sequence of complex number bounded by $1$ satisfying
$\sum\mathrm{Re}\,a_{i}=\infty$. ∎

By appealing to E, we first prove C in the case when $f$ takes values in the
unit circle.

######  Lemma 4.6.

Let $f:\mathbb{G}^{\ast}\to\mathbb{C}$ be a completely multiplicative function
taking values in the unit circle such that $f(\mathrm{i})=1$ and
$f(\mathbb{P})$ is finite. Then $\mathop{\mathbb{E}}(f)$ exists and equals
$P(f)$.

###### Proof.

Since $f(\mathbb{P})$ is finite, $f(\mathbb{P}_{1})$ is also finite. If the
set $\\{p\in\mathbb{P}_{1}:f(p)\neq 1\\}$ is convergent, then the conclusion
follows immediately from applying Lemma 4.3 with $g\equiv 1$. Otherwise,
because $f(\mathbb{P}_{1})$ is finite, it follows from Lemma 4.5 that
$P(f)=0$. Therefore, our goal is to show that, when the set
$\\{p\in\mathbb{P}_{1}:f(p)\neq 1\\}$ is divergent, $\mathop{\mathbb{E}}(f)$
exists and equals $0$. We do this by invoking E.

Let $X_{0}$ be the (finite) set of those points $z\in S^{1}$ in the unit
circle for which the set $\\{p\in\mathbb{P}_{1}:f(p)=z\\}$ is divergent and
let $X$ be the closed subgroup of $S^{1}$ generated by $X_{0}$. Using the
assumption that the set $\\{p\in\mathbb{P}_{1}:f(p)\neq 1\\}$ is divergent, we
deduce that $X$ is not the singleton $\\{1\\}$. We “remove” the exceptional
primes by considering the completely multiplicative function
$g:\mathbb{G}^{*}\to\mathbb{C}$ defined by $g(p)=f(p)$ for any prime
$p\in\mathbb{P}_{1}$ with $f(p)\in X_{0}$, $g(p)=1$ for any other prime $p$,
and $g(\mathrm{i})=1$. Since $f(\mathbb{P}_{1})$ is finite, it follows that
$f$ and $g$ only differ on a convergent set of primes and hence, using again
Lemma 4.3, it suffices to show that $\mathop{\mathbb{E}}(g)$ exists and equals
$0$.

Since $X$ is the closed group generated by $g(\mathbb{P}_{1})$, the Haar
measure $\mu$ on $X$ is the unique measure on $X$ preserved under all the maps
$\tau(n):x\mapsto g(n)x$ for $n\in\mathbb{G}^{*}$. Letting $F:X\to\mathbb{C}$
be the identity function and $x=1$, it follows from E that
$\mathop{\mathbb{E}}(g)$ exists and equals
$\displaystyle\int_{X}z\,\mathrm{d}\mu(z)$. Since $\mu$ is the Haar measure on
a subgroup of $S^{1}$ other than the trivial group $\\{1\\}$, the integral is
$0$, finishing the proof. ∎

We are now ready to prove C.

###### Proof of C.

Let $f:\mathbb{G}^{\ast}\to\mathbb{C}$ be a bounded completely multiplicative
such that the set $\operatorname{Arg}(f(\mathbb{P}))$ is finite.

If
$\displaystyle\sum_{p\in\mathbb{P}_{1}}\frac{1-|f(p)|}{\mathcal{N}(p)}=\infty$,
then by Corollary 4.4 we have $\mathop{\mathbb{E}}(|f|)=0$ which implies that
$\mathop{\mathbb{E}}(f)=0$. Since $|1-f(p)|\geq 1-|f(p)|$, we also have
$\displaystyle\sum_{p\in\mathbb{P}_{1}}\frac{|1-f(p)|}{\mathcal{N}(p)}=\infty$
and hence by Lemma 4.5, $P(f)=0$ as well.

If
$\displaystyle\sum_{p\in\mathbb{P}_{1}}\frac{1-|f(p)|}{\mathcal{N}(p)}<\infty$,
then we can approximate $f$ by the completely multiplicative function taking
values in the unit circle $g(n):=e^{\mathrm{i}\operatorname{Arg}(f(n))}$.
Indeed,

$\sum_{p\in\mathbb{P}_{1}}\frac{|g(p)-f(p)|}{\mathcal{N}(p)}=\sum_{p\in\mathbb{P}_{1}}\frac{|1-f(p)/g(p)|}{\mathcal{N}(p)}=\sum_{p\in\mathbb{P}_{1}}\frac{1-|f(p)|}{\mathcal{N}(p)}<\infty.$
(4.5)

Moreover, since $\operatorname{Arg}(f(\mathbb{P}_{1}))$ is finite, the range
$g(\mathbb{P}_{1})$ is finite, and so Lemma 4.6 implies that
$\mathop{\mathbb{E}}(g)$ exists and equals $P(g)$. By Lemma 4.3,
$\mathop{\mathbb{E}}(f)$ also exists and equals $P(f)$.

∎

From C, one can easily deduce an analogous result for arbitrary bounded
completely multiplicative function $f$ without the restriction that
$f(\mathrm{i})=1$. Let
$Q_{1}=\\{z\in\mathbb{G}:\mathrm{Re}\,(z)>0,\mathrm{Im}\,(z)\geq 0\\}$ denote
the first quadrant of the complex plane $\mathbb{C}$. Let
$Q_{2}=\mathrm{i}Q_{1},Q_{3}=\mathrm{i}^{2}Q_{1}$ and
$Q_{4}=\mathrm{i}^{3}Q_{1}$ be the other quadrants. Let $\operatorname{m}$ be
the Lebesgue measure on $\mathbb{C}$ and note that since $f$ is completely
multiplicative, $f(\mathrm{i})$ must be either $1,-1,\mathrm{i}$ or
$-\mathrm{i}$.

######  Theorem 4.7.

Let $f:\mathbb{G}^{*}\to\mathbb{C}$ be a bounded completely multiplicative
function such that $\operatorname{Arg}(f(\mathbb{P}))$ is finite. If
$(\Phi_{N})_{N\in\mathbb{N}}$ is a dilated Følner sequence corresponding to
the Jordan measurable set $U$, then the limit
$\lim_{N\to\infty}\mathop{\mathbb{E}}_{n\in\Phi_{N}}f(n)$ exists and equals

$P(f)\cdot\sum_{k=0}^{3}\frac{\operatorname{m}(U\cap
Q_{k+1})}{\operatorname{m}(U)}\cdot f(\mathrm{i})^{k}.$

###### Proof.

If $(\Phi_{N})_{N\in\mathbb{N}}$ is a dilated Følner sequence in the first
quadrant $Q_{1}$, the value of $f(\mathrm{i})$ does not affect the average
$\mathop{\mathbb{E}}_{n\in\Phi_{N}}f(n)$. Therefore, in this case, C implies
that

$\lim_{N\to\infty}\mathop{\mathbb{E}}_{n\in\Phi_{N}}f(n)=P(f).$

Now, let $(\Phi_{N})_{N\in\mathbb{N}}$ be an arbitrary dilated Følner sequence
in $\mathbb{G}$ and $U$ be a Jordan measurable set associated to
$(\Phi_{N})_{N\in\mathbb{N}}$. For $k\in\\{0,1,2,3\\}$, we have

$\mathop{\mathbb{E}}_{n\in\Phi_{N}\cap
Q_{k+1}}f(n)=\mathop{\mathbb{E}}_{n\in\Phi_{N}\cap
Q_{k+1}}f(\mathrm{i}^{k})f(-\mathrm{i}^{k}n)=f(\mathrm{i})^{k}\mathop{\mathbb{E}}_{n\in-\mathrm{i}^{k}(\Phi_{N}\cap
Q_{k+1})}f(n)$

which converges to $f(\mathrm{i})^{k}P(f)$ as $N\to\infty$. Here we use the
fact that $(-\mathrm{i}^{k}(\Phi_{N}\cap Q_{k+1}))_{N\in\mathbb{N}}$ is a
dilated Følner sequence in the first quadrant. Lastly, observe that

$\lim_{N\to\infty}\mathop{\mathbb{E}}_{n\in\Phi_{N}}f(n)=\sum_{k=0}^{3}\frac{\operatorname{m}(U\cap
Q_{k+1})}{\operatorname{m}(U)}\cdot\lim_{N\to\infty}\mathop{\mathbb{E}}_{n\in\Phi_{N}\cap
Q_{k+1}}f(n)$

and so our theorem follows. ∎

###  4.3. Proofs of Theorems A, 1.2 and B

The implications C $\Rightarrow$ B $\Rightarrow$ Theorem 1.2 are easy and have
been outlined in the introduction. To prove A, given a bounded completely
multiplicative function $f:\mathbb{N}\to\mathbb{R}$, we apply Theorem 1.2 to
the function $f\circ\mathcal{N}:\mathbb{G}^{*}\to\mathbb{R}$. As mentioned in
Section 2.1, a Gaussian integer $a+b\mathrm{i}$ is in $\mathbb{P}_{1}$ if and
only if either:

  * •

$b=0$ and $a=p$ is a prime in $\mathbb{N}$ of the form $4n+3$, or

  * •

$b>0$ and $a^{2}+b^{2}=p$ is a prime number in $\mathbb{N}$ (which will not be
of the form $4n+3)$.

In the first case, the norm of the Gaussian prime $a+b\mathrm{i}$ is $p^{2}$
and in the second case, the norm is $p$. For each integer prime $p$ of the
form $4n+1$, there are two Gaussian primes in $\mathbb{P}_{1}$ associated to
it: $a+b\mathrm{i}$ and $b+a\mathrm{i}$. Now it is simple to check that
$P(f\circ\mathcal{N})$ equals the expression in (1.5).

###  4.4. Proof of D

Suppose $(X,T)$ is a uniquely ergodic system with the unique invariant measure
$\mu$. Denote by ${\mathcal{T}}$ the semigroup of all continuous self-maps of
$X$. For $n\in\mathbb{G}^{\ast}$, define

$\tau(n)=T^{\Omega(\mathcal{N}(n))}.$

Then $\tau:(\mathbb{G}^{*},\times)\to{\mathcal{T}}$ is a semigroup
homomorphism because

$\tau(mn)=T^{\Omega(\mathcal{N}(mn))}=T^{\Omega(\mathcal{N}(m)\mathcal{N}(n))}=T^{\Omega(\mathcal{N}(m))+\Omega(\mathcal{N}(n))}=\tau(m)\circ\tau(n).$

Gaussian primes $p\in\mathbb{P}$ fall into two categories: either
$\mathcal{N}(p)$ is a prime in $\mathbb{Z}$ which is $\equiv 1\bmod 4$ or
$\mathcal{N}(p)$ is the square of a prime in $\mathbb{Z}$ which is $\equiv
3\bmod 4$. The latter category forms a convergent set of Gaussian primes
(those that are either real or purely imaginary), so for all $p\in\mathbb{P}$
outside a convergent set we have that $\mathcal{N}(p)$ is an integer prime,
and hence $\tau(p)=T$. Therefore, by E, for any $x\in X$ and $F\in C(X)$,

$\lim_{N\to\infty}\frac{1}{N^{2}}\sum_{m,n=1}^{N}F(T^{\Omega(m^{2}+n^{2})}x)=\lim_{N\to\infty}\frac{1}{N^{2}}\sum_{z\in\Phi_{N}}F(\tau(z)x)=\int_{X}F\
d\mu,$

where $(\Phi_{N})_{N\in\mathbb{N}}$ is the dilated Følner sequence given by
$\Phi_{N}:=\\{z\in\mathbb{G}^{\ast}\colon 0<\mathrm{Re}\,z,\mathrm{Im}\,z\leq
N\\}$.

##  5\. Open questions

As mentioned in the introduction, one of the main motivations for this paper
was the question of Frantzikinakis and Host, 1.1, which roughly speaking asks
whether the Cesàro average of a real valued bounded completely multiplicative
function over a homogeneous polynomial of two variables must always exist. Our
A gives a positive answer for the specific polynomial $P(m,n)=m^{2}+n^{2}$,
and while our approach might be adaptable to handle other norm forms, the case
of general polynomials remains out of reach.

In a similar vein, using D as motivation, we ask whether the polynomial
$P(m,n)=m^{2}+n^{2}$ can be replaced with more general polynomials:

######  Question 5.1.

Let $(X,T)$ be a uniquely ergodic system. Let $P\in\mathbb{Z}[x,y]$ be a
homogeneous polynomial taking values on the positive integers and let $F\in
C(X)$. Does the limit

$\lim_{N\to\infty}\mathop{\mathbb{E}}_{1\leq m,n\leq N}F(T^{\Omega(P(m,n))}x)$

exist?

In [22], Loyd showed that an analogue of Birkhoff pointwise ergodic theorem
along $\Omega(n)$ is false. More precisely, in every non-atomic ergodic system
$(X,\mu,T)$, there is a measurable set $A\subset X$ such that for $\mu$-almost
every $x\in X$,

$\limsup_{N\to\infty}\mathop{\mathbb{E}}_{1\leq n\leq
N}1_{A}(T^{\Omega(n)}x)=1\text{ and
}\liminf_{N\to\infty}\mathop{\mathbb{E}}_{1\leq n\leq
N}1_{A}(T^{\Omega(n)}x)=0.$

In light of D, we can ask a similar question regarding the averages along
$\Omega(m^{2}+n^{2})$:

######  Question 5.2.

Let $(X,\mu,T)$ be a non-atomic ergodic system. Is it true that there exists a
measurable set $A\subset X$ such that for $\mu$-almost every $x\in X$,

$\limsup_{N\to\infty}\mathop{\mathbb{E}}_{1\leq m,n\leq
N}1_{A}(T^{\Omega(m^{2}+n^{2})}x)=1\text{ and
}\liminf_{N\to\infty}\mathop{\mathbb{E}}_{1\leq m,n\leq
N}1_{A}(T^{\Omega(m^{2}+n^{2})}x)=0?$

In the same paper, Loyd [22] proved that Bergelson and Richter’s result
(Theorem 1.5) would be false in general if we removed the unique ergodicity
assumption. It was shown that there exists an ergodic system $(X,\mu,T)$, a
$\mu$-generic point $x\in X$ and a continuous function $F\in C(X)$ such that

$\lim_{N\to\infty}\mathop{\mathbb{E}}_{n\in[N]}F(T^{\Omega(n)}x)$

does not exist. This result suggests the following question:

######  Question 5.3.

Does there exist an ergodic system $(X,\mu,T)$, a $\mu$-generic point $x\in X$
and a continuous function $F\in C(X)$ such that

$\mathop{\mathbb{E}}_{1\leq m,n\leq N}F(T^{\Omega(m^{2}+n^{2})}x)$

does not converge as $N\to\infty$?

Theorems B and C are true for dilated Følner sequences. However, as discussed
in Section 2.3 and also Appendix B, the conclusions of these theorems do not
hold for every additive Følner sequence $(\Phi_{N})$ in $\mathbb{G}$. This
raises the following open-ended question:

######  Question 5.4.

Which additive Følner sequences $(\Phi_{N})_{N\in\mathbb{N}}$ in
$\mathbb{G}^{\ast}$ are such that for every bounded real-valued completely
multiplicative function $f:\mathbb{G}^{\ast}\to\mathbb{R}$, the averages
$\mathop{\mathbb{E}}_{n\in\Phi_{N}}f(n)$ converge?

The sequence
$\Phi_{N}=N+\\{n\in\mathbb{G}:|\mathrm{Re}\,n|,|\mathrm{Im}\,n|<N\\}$ is a
dilated Følner sequence corresponding to the open set
$U=\\{z\in\mathbb{G}:|\mathrm{Re}\,z-1|,|\mathrm{Im}\,z|<1\\}$. On the other
hand, if we shift each square by $N^{1+\varepsilon}$ we destroy this property.
Thus, a concrete instance of 5.4 is following:

######  Question 5.5.

Is there $\varepsilon>0$ such that, letting
$\Phi_{N}=N^{1+\varepsilon}+\\{n\in\mathbb{G}:|\mathrm{Re}\,n|,|\mathrm{Im}\,n|<N\\}$,
for every bounded completely multiplicative function
$f:\mathbb{G}^{*}\to\mathbb{R}$ the average
$\mathop{\mathbb{E}}_{n\in\Phi_{N}}f(n)$ converges?

In a similar direction, and in light of B and a recent result on averages of
multiplicative functions on short intervals of Matomäki and Radziwiłł [23], we
ask the following question:

######  Question 5.6.

Let $f:\mathbb{G}^{*}\to\mathbb{R}$ be a bounded completely multiplicative
function and let $\Phi_{N}=\\{n:|\mathrm{Re}\,n|,|\mathrm{Im}\,n|\leq N\\}$.
Define $P(f)$ as in (1.6). Is it true that

$\lim_{H\to\infty}\lim_{N\to\infty}\mathop{\mathbb{E}}_{n\in\Phi_{N}}\left|\mathop{\mathbb{E}}_{h\in
n+\Phi_{H}}f(h)-P(f)\right|=0?$

More generally, does this hold for any dilated Følner sequence
$(\Phi_{N})_{N\in\mathbb{N}}$?

In the last three questions, it also makes sense to ask whether the
restriction that the completely multiplicative functions involved are real
valued can be relaxed to allow completely multiplicative functions
$f:\mathbb{G}^{*}\to\mathbb{C}$ satisfying
$\operatorname{Arg}(f(\mathbb{P}))<\infty$.

##  Appendix A Some estimates

######  Lemma A.1 (Turán-Kubilius).

Let $(\Phi_{N})_{N\in\mathbb{N}}$ be a Følner sequence in $(\mathbb{G},+)$ and
let $B\subset\mathbb{G}^{\ast}$ be finite and non-empty. For any function
$a\colon\mathbb{G}\to\mathbb{C}$ bounded by 1, we have that

$\limsup_{N\to\infty}\left|\mathop{\mathbb{E}}_{n\in\Phi_{N}}a(n)-\mathop{\mathbb{E}}_{p\in
B}^{\log}\mathop{\mathbb{E}}_{n\in\Phi_{N}/p}a(pn)\right|\leq\left(\mathop{\mathbb{E}}_{p\in
B}^{\log}\mathop{\mathbb{E}}_{q\in
B}^{\log}\mathcal{N}\big{(}\gcd(p,q)\big{)}-1\right)^{1/2}.$

###### Proof.

Denote by $d(A):=\lim_{N\to\infty}\mathop{\mathbb{E}}_{n\in\Phi_{N}}1_{A}(n)$
when the limit exists. Note that, for every $p,q\in\mathbb{G}^{\ast}$,

$d(p\mathbb{G})=\frac{1}{\mathcal{N}(p)}\qquad\text{ and }\qquad
d(p\mathbb{G}\cap
q\mathbb{G})=\frac{\mathcal{N}(\gcd(p,q))}{\mathcal{N}(p)\mathcal{N}(q)}.$

For any finite set $B\subset\mathbb{G}^{\ast}$ and any $n\in\mathbb{G}$,

$\displaystyle\left|\mathop{\mathbb{E}}^{\log}_{p\in B}1-\mathcal{N}(p)\cdot
1_{p\mathbb{G}}(n)\right|^{2}$ $\displaystyle=$
$\displaystyle\mathop{\mathbb{E}}^{\log}_{p\in
B}\mathop{\mathbb{E}}^{\log}_{q\in B}(1-\mathcal{N}(p)\cdot
1_{p\mathbb{G}}(n))(1-\mathcal{N}(q)\cdot 1_{q\mathbb{G}}(n))$
$\displaystyle=$ $\displaystyle\mathop{\mathbb{E}}^{\log}_{p\in
B}\mathop{\mathbb{E}}^{\log}_{q\in
B}1+\mathcal{N}(p)\mathcal{N}(q)1_{p\mathbb{G}\cap
q\mathbb{G}}(n)-\mathcal{N}(p)1_{p\mathbb{G}}(n)-\mathcal{N}(q)1_{q\mathbb{G}}(n)$

so averaging over $n$ we deduce that

$\lim_{N\to\infty}\mathop{\mathbb{E}}_{n\in\Phi_{N}}\left|\mathop{\mathbb{E}}^{\log}_{p\in
B}1-\mathcal{N}(p)\cdot
1_{p\mathbb{G}}(n)\right|^{2}=\mathop{\mathbb{E}}^{\log}_{p\in
B}\mathop{\mathbb{E}}^{\log}_{q\in B}\mathcal{N}(\gcd(p,q))-1$

An application of the Cauchy-Schwarz inequality then yields the estimate

$\limsup_{N\to\infty}\left|\mathop{\mathbb{E}}_{n\in\Phi_{N}}a(n)-\mathop{\mathbb{E}}_{p\in
B}^{\log}\mathcal{N}(p)\mathop{\mathbb{E}}_{n\in\Phi_{N}}a(n)1_{p\mathbb{G}}(n)\right|\leq\left(\mathop{\mathbb{E}}_{p\in
B}^{\log}\mathop{\mathbb{E}}_{q\in
B}^{\log}\mathcal{N}\big{(}\gcd(p,q)\big{)}-1\right)^{1/2}.$ (A.1)

Finally notice that

$\mathop{\mathbb{E}}_{n\in\Phi_{N}}a(n)1_{p\mathbb{G}}(n)=\frac{1}{|\Phi_{N}|}\sum_{n\in\Phi_{N}\cap
p\mathbb{G}}a(n)=\frac{|\Phi_{N}\cap
p\mathbb{G}|}{|\Phi_{N}|}\mathop{\mathbb{E}}_{n\in\Phi_{N}/p}a(np),$

and since $d(p\mathbb{G})=1/\mathcal{N}(p)$, (A.1) is equivalent to the
desired conclusion. ∎

To achieve the last condition in Lemma 3.2 we use the following estimates.

######  Lemma A.2.

Let $F_{1},F_{2}\subset\mathbb{P}$ be finite sets of primes. Then

  1. (i)

$\displaystyle\mathop{\mathbb{E}}_{n,m\in
F_{1}}^{\log}\mathcal{N}(\gcd(n,m))<1+\frac{4}{\log(F_{1})}$.

  2. (ii)

$\displaystyle\mathop{\mathbb{E}}_{n,m\in
F_{1}F_{2}}^{\log}\mathcal{N}(\gcd(n,m))<\left(1+\frac{8}{\log(F_{1})}\right)\left(1+\frac{8}{\log(F_{2})}\right)$.

###### Proof.

  1. (i)

Denote by $U=\\{1,i,-1,-i\\}$ the group of units on $\mathbb{G}$. Note that
for $m,n\in\mathbb{P}$,

$\mathcal{N}(\gcd(m,n))=\begin{cases}\mathcal{N}(m)\text{ if }m\in nU\\\
1\text{ otherwise.}\end{cases}$

Therefore, for every $n\in F_{1}$,

$\mathop{\mathbb{E}}^{\log}_{m\in
F_{1}}\mathcal{N}(\gcd(n,m))=\frac{1}{\log(F_{1})}\sum_{m\in
F_{1}}\frac{\mathcal{N}(\gcd(n,m))}{\mathcal{N}(m)}\leq\frac{1}{\log(F_{1})}\left(4+\sum_{m\in
F_{1}}\frac{1}{\mathcal{N}(m)}\right)=1+\frac{4}{\log(F_{1})}$

Averaging in $n\in F_{1}$ yields the desired result.

  2. (ii)

Given $n,m\in F_{1}F_{2}$ we can decompose $n=n_{1}n_{2}$ and $m=m_{1}m_{2}$
with $n_{i},m_{i}\in F_{i}$. Note that

$\mathcal{N}(\gcd(n,m))=\begin{cases}\mathcal{N}(m)&\text{if
}\\{m_{1},m_{2}\\}\subset n_{1}U\cup n_{2}U\\\ \mathcal{N}(m_{i})&\text{if
}m_{i}\in n_{1}U\cup n_{2}U\text{ and }m_{3-i}\notin n_{1}U\cup n_{2}U\\\
1&\text{otherwise}\end{cases}$

For a fixed $n\in F_{1}F_{2}$, partition $F_{1}F_{2}$ into the following $3$
sets:

$A_{1}=\\{m\in F_{1}F_{2}:\mathcal{N}(\gcd(n,m))=\mathcal{N}(m)\\},\qquad
A_{2}:=\\{m\in F_{1}F_{2}:\mathcal{N}(\gcd(n,m))=\mathcal{N}(m_{i})\\},$
$A_{3}:=\\{m\in F_{1}F_{2}:\mathcal{N}(\gcd(n,m))=1\\}.$

We can then estimate the sum

$\displaystyle\sum_{m\in
F_{1}F_{2}}\frac{\mathcal{N}(\gcd(m,n))}{\mathcal{N}(m)}$ $\displaystyle=$
$\displaystyle\left(\sum_{m\in A_{1}}+\sum_{m\in A_{2}}+\sum_{m\in
A_{3}}\right)\frac{\mathcal{N}(\gcd(m,n))}{\mathcal{N}(m)}$
$\displaystyle\leq$ $\displaystyle
64+8\big{(}\log(F_{1})+\log(F_{2})\big{)}+\log(F_{1}F_{2}).$

Dividing by $\log(F_{1}F_{2})=\log(F_{1})\log(F_{2})$ we conclude that for
each $n\in F_{1}F_{2}$ we have

$\mathop{\mathbb{E}}_{m\in
F_{1}F_{2}}^{\log}\mathcal{N}(\gcd(m,n))\leq\left(1+\frac{8}{\log(F_{1})}\right)\left(1+\frac{8}{\log(F_{2})}\right).$

Averaging over $n\in F_{1}F_{2}$ yields the desired result.

∎

The following estimate is needed in the main text; we include its short proof
for completeness.

######  Lemma A.3.

Let $F$ be a non-empty finite set and let $v,w:F\to\mathbb{R}^{>0}$ and
$f,g:F\to\mathbb{C}$. Suppose that $|f(p)|\leq 1$ for all $p\in F$ and that
there exists $\delta>0$ such that for each $p\in F$,

$\left|\frac{w(p)}{v(p)}-1\right|<\delta,\qquad\text{ and
}\qquad\big{|}f(p)-g(p)\big{|}<\delta.$

Then

$\left|\frac{\sum_{p\in F}w(p)f(p)}{\sum_{p\in F}w(p)}-\frac{\sum_{p\in
F}v(p)g(p)}{\sum_{p\in F}v(p)}\right|<3\delta.$

###### Proof.

All sums in the proof are over $p\in F$, so we will omit this subscript. First
note that

$\left|\frac{\sum v(p)g(p)}{\sum v(p)}-\frac{\sum_{p\in F}v(p)f(p)}{\sum
v(p)}\right|\leq\frac{1}{\sum v(p)}\sum v(p)\big{|}g(p)-f(p)\big{|}<\delta.$
(A.2)

Second we have $\big{|}w(p)-v(p)\big{|}\leq\delta v(p)$ and hence

$\left|\frac{\sum v(p)f(p)}{\sum v(p)}-\frac{\sum w(p)f(p)}{\sum
v(p)}\right|=\frac{\left|\sum\big{(}v(p)-w(p)\big{)}f(p)\right|}{\sum
v(p)}\leq\frac{\sum\delta v(p)\big{|}f(p)\big{|}}{\sum v(p)}\leq\delta.$ (A.3)

Finally we have

$\displaystyle\left|\frac{\sum w(p)f(p)}{\sum w(p)}-\frac{\sum w(p)f(p)}{\sum
v(p)}\right|$ $\displaystyle=$ $\displaystyle\left|\sum
w(p)f(p)\right|\cdot\left|\frac{1}{\sum w(p)}-\frac{1}{\sum v(p)}\right|$
$\displaystyle\leq$ $\displaystyle\left(\sum
w(p)\right)\cdot\left|\frac{1}{\sum w(p)}-\frac{1}{\sum v(p)}\right|$
$\displaystyle=$ $\displaystyle\frac{\left|\sum v(p)-w(p)\right|}{\sum
v(p)}\leq\frac{\delta\sum v(p)}{\sum v(p)}=\delta.$

Together with (A.2) and (A.3) this yields the desired conclusion. ∎

##  Appendix B A counterexample with non-dilated Følner sequences

A function $f:\mathbb{G}^{\ast}\to\\{-1,1\\}$ is called _normal_ if every
finite pattern of $-1$ and $1$ appears in $f$ at the correct frequency; that
is for all $k\in\mathbb{N}$, all distinct $h_{1},\ldots,h_{k}\in\mathbb{G}$,
and $\varepsilon_{1},\ldots,\varepsilon_{k}\in\\{-1,1\\}$, the set

$S:=\big{\\{}n\in\mathbb{G}:f(n+h_{j})=\varepsilon_{j}\text{ for all
}j=1,\ldots,k\big{\\}}$

satisfies

$\lim_{N\to\infty}\frac{|S\cap N\mathbb{D}|}{|\mathbb{G}\cap
N\mathbb{D}|}=\frac{1}{2^{k}},$

where we write $f(0)=0$ for convenience. In this section, to demonstrate the
necessity of dilated Følner sequences in Theorems B, C, and E, we show that a
“random” completely multiplicative function $f:\mathbb{G}^{\ast}\to\\{-1,1\\}$
is almost surely normal, which implies that along some additive Følner
sequences it is indistinguishable from the constant $1$ sequence. The
analogous result for multiplicative functions on $\mathbb{Z}$ was proved by
Fish [10] and that proof extends to the Gaussian integer setting without major
difficulties; we provide a proof here for completeness.

######  Lemma B.1 ([10, Lemma 2.2]).

Let $(a_{n})_{n\in\mathbb{N}}$ be a bounded sequence. Let
$T_{N}=\frac{1}{N}\sum_{n=1}^{N}a_{n}$ and $t\in\mathbb{C}$. The followings
are equivalent:

  1. (1)

$\lim_{N\to\infty}T_{N}=t$,

  2. (2)

There exists a sequence of increasing indices $(N_{j})_{j\in\mathbb{N}}$ such
that $N_{j+1}/N_{j}\to 1$ and $\lim_{j\to\infty}T_{N_{j}}=t$.

######  Proposition B.2.

Define a random completely multiplicative function
$f:\mathbb{G}^{\ast}\to\\{-1,1\\}$ as follows: let $f(i)=1$, for every
$p\in\mathbb{P}$, let $f(p)=1$ or $-1$ with probability $1/2$ each. For
convenience write $f(0)=0$. Then almost surely,

$\lim_{N\to\infty}\mathop{\mathbb{E}}_{\begin{subarray}{c}n\in\mathbb{G}\\\
\mathcal{N}(n)<N\end{subarray}}f(n+h_{1})\cdots f(n+h_{k})=0$

for all $k\in\mathbb{N}$ and pairwise distinct
$h_{1},\ldots,h_{k}\in\mathbb{G}$.

###### Proof.

In this proof, we use $\mathop{\mathbb{E}}$ for averages and $E$ for expected
values of random variables.

Fix $k$ and distinct $h_{1},\ldots,h_{k}\in\mathbb{G}$. For $n\in\mathbb{G}$,
define

$\xi(n):=(n+h_{1})\cdots(n+h_{k})$

and

$\phi(n):=f(\xi(n)),$

and for $N\in\mathbb{N}$, define

$T_{N}=\mathop{\mathbb{E}}_{\begin{subarray}{c}n\in\mathbb{G}\\\
\mathcal{N}(n)<N\end{subarray}}\phi(n).$

Our goal is to show $T_{N}\to 0$ almost surely. In order to do this, we will
show

$\sum_{N=1}^{\infty}E(T_{N^{40}}^{2})<\infty.$

Note that

$T_{N}^{2}=\mathop{\mathbb{E}}_{\begin{subarray}{c}x,y\in\mathbb{G}\\\
\mathcal{N}(x)<N\\\ \mathcal{N}(y)<N\end{subarray}}\phi(x)\phi(y).$

Therefore, by the linearity of expectation, we get

$E(T_{N}^{2})=\mathop{\mathbb{E}}_{\begin{subarray}{c}x,y\in\mathbb{G}\\\
\mathcal{N}(x)<N\\\ \mathcal{N}(y)<N\end{subarray}}E(\phi(x)\phi(y)).$

We have that

$E(\phi(x)\phi(y))=\begin{cases}1\text{ if }\xi(x)\xi(y)\text{ is a square}\\\
0\text{ otherwise}.\end{cases}$

Thus, in order to bound $E(T_{N}^{2})$, we need to bound the number of pairs
$(x,y)\in B_{N}(0)$ such that $\xi(x)\xi(y)$ is a square.

For any $x\in\mathbb{G}^{\ast}\cap N\mathbb{D}$, write
$\xi(x)=q_{1}q_{2}\cdots q_{\ell}m^{2}$ where $q_{1},q_{2},\ldots,q_{\ell}$
are distinct primes. Define $h(x)=\ell$, the number of prime divisors of the
square-free component of $\xi(x)$. Let $D$ be the set of all possible Gaussian
integers that divide at least two numbers in $\\{n+h_{1},\ldots,n+h_{k}\\}$.
Thus $D$ is the set of all possible divisors of $h_{j_{1}}-h_{j_{2}}$ for
$1\leq j_{1}<j_{2}\leq k$. For a finite subset $S\subset\mathbb{G}$, denote by
$m(S)$ the product of all elements in $S$ and define $m(\varnothing)=1$.

With the above $x$, if $\xi(x)\xi(y)$ is a square, there exists $S_{1}\subset
D$ and $S_{2}\subset\\{q_{1},\ldots,q_{\ell}\\}$ such that
$y=m(S_{1})m(S_{2})n^{2}$ for some $n\in\mathbb{G}$. Thus, the number of $y\in
B_{N}(0)$ such that $\xi(x)\xi(y)$ is a square is at most

$2^{|D|}\cdot 2^{h(x)}|B_{\sqrt{N}}(0)|\ll 2^{h(x)}\sqrt{N}.$

Thus

$E(T_{N}^{2})\ll\frac{1}{N^{3/2}}\sum_{x\in\mathbb{G}\cap
N\mathbb{D}}2^{h(x)}.$

For any positive integer $M$, if all prime divisors of $\xi(x)$ has norms
greater than $M$, then

$h(x)<\log_{M}(\mathcal{N}(x+h_{k}))^{k}=k\log_{M}\mathcal{N}(x+h_{k}).$

Fix an $M$ such that $M>2^{k/0.45}$ and let $C$ be the number of primes whose
norms do not exceed $M$. Then for any $x\in\mathbb{G}$,

$h(x)<C+k\log_{M}\mathcal{N}(x+h_{k}).$

It follows that

$2^{h(x)}<2^{C}\cdot
2^{k\log_{M}\mathcal{N}(x+h_{k})}<2^{C}\cdot\mathcal{N}(x+h_{k})^{0.45}.$

Therefore,

$E(T_{N}^{2})\ll\frac{1}{N^{3/2}}\sum_{x\in
N\mathbb{D}}\mathcal{N}(x+h_{k})^{0.45}\ll\frac{1}{N^{0.05}},$

and so

$\sum_{N=1}^{\infty}E(T_{N^{40}}^{2})\ll\sum_{n=1}^{\infty}\frac{1}{N^{2}}<\infty.$

Thus $T_{N^{40}}\to 0$ almost surely and by Lemma B.1, $T_{N}\to 0$ almost
surely. ∎

The next lemma is about the equivalence of Chowla’s conjecture and the
normality of the Liouville function. This result is well-known in $\mathbb{Z}$
and its proof in $\mathbb{G}$ is the same. However, since we could not locate
the exact proof in the literature, we include it for completeness.

######  Lemma B.3.

If $f:\mathbb{G}\to\\{-1,1\\}$ satisfies

$\lim_{N\to\infty}\mathop{\mathbb{E}}_{\begin{subarray}{c}n\in\mathbb{G}\\\
\mathcal{N}(n)<N\end{subarray}}f(n+h_{1})\cdots f(n+h_{k})=0$ (B.1)

for every $k\in\mathbb{N}$ and distinct $h_{1},\ldots,h_{k}\in\mathbb{G}$,
then $f$ is normal.

###### Proof.

Let $X=\\{-1,1\\}^{\mathbb{G}}$ and $T$ be the $\mathbb{G}$-action on $X$
defined by $T_{g}(x(n))=x(n+g)$ for all $g\in\mathbb{G}$ and $x\in X$. Let
$\mu$ be a weak∗-limit of the sequence of measures

$\mathop{\mathbb{E}}_{\begin{subarray}{c}g\in\mathbb{G}\\\
\mathcal{N}(g)<N\end{subarray}}\delta_{T_{g}f}$

where $\delta_{T_{g}f}$ is the Dirac measure at $T_{g}f$. Define a function
$F:X\to\\{-1,1\\}$ by $F(x)=x(0)$ for all $x\in\mathbb{G}$. Then by (B.1) and
the definition of $\mu$, for all $k\in\mathbb{N}$ and distinct
$h_{1},\ldots,h_{k}\in\mathbb{G}$,

$\int_{X}T^{h_{1}}F\cdots T^{h_{k}}F\ d\mu=0=\left(\int_{X}F\
d\mu\right)^{k}.$

Since $C(X)$ contains a dense subset which is generated by the family of
functions of the form $T^{h_{1}}F\cdots T^{h_{k}}F$, we deduce that $\mu$ is
the Bernoulli measure on $X$. Thus for any $k\in\mathbb{N}$, distinct
$h_{1},\ldots,h_{k}\in\mathbb{G}$, and
$\varepsilon_{1},\ldots,\varepsilon_{k}\in\\{-1,1\\}$, the cylinder set

$C:=\\{x\in X:x(h_{j})=\varepsilon_{j}\text{ for all }j\in[k]\\}$

has measure $\mu(C)=1/2^{k}$. By the definition of $\mu$, it means that that
pattern of $-1,1$ appears in $f$ at the correct frequency. ∎

Proposition B.2 and Lemma B.3 imply the proposition:

######  Proposition B.4.

Define a random completely multiplicative function
$f:\mathbb{G}^{\ast}\to\\{-1,1\\}$ as follows: let $f(\mathrm{i})=1$ and, for
every prime $p\in\mathbb{P}_{1}$, let $f(p)=1$ or $-1$ with probability $1/2$
each. Then almost surely, $f$ is normal.

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