# Projection constants for spaces of Dirichlet polynomials

A. Defant  Institut für Mathematik, Carl von Ossietzky Universität, 26111
Oldenburg, Germany<EMAIL_ADDRESS>,  D. Galicer
Departamento de Matemática, Facultad de Cs. Exactas y Naturales, Universidad
de Buenos Aires and IMAS-CONICET. Ciudad Universitaria, Pabellón I (C1428EGA)
C.A.B.A., Argentina<EMAIL_ADDRESS>,  M. Mansilla  Departamento de
Matemática, Facultad de Cs. Exactas y Naturales, Universidad de Buenos Aires
and IAM-CONICET. Saavedra 15 (C1083ACA) C.A.B.A., Argentina
<EMAIL_ADDRESS>,  M. Mastyło  Faculty of Mathematics and Computer
Science, Adam Mickiewicz University, Poznań, Uniwersytetu Poznańskiego 4,
61-614 Poznań, Poland<EMAIL_ADDRESS>and  S. Muro  FCEIA,
Universidad Nacional de Rosario and CIFASIS, CONICET, Ocampo $\&$ Esmeralda,
S2000 Rosario, Argentina<EMAIL_ADDRESS>
###### Abstract.

Given a frequency sequence $\omega=(\omega_{n})$ and a finite subset
$J\subset\mathbb{N}$, we study the space $\mathcal{H}_{\infty}^{J}(\omega)$ of
all Dirichlet polynomials $D(s):=\sum_{n\in
J}a_{n}e^{-\omega_{n}s},\,s\in\mathbb{C}$. The main aim is to prove
asymptotically correct estimates for the projection constant
$\boldsymbol{\lambda}\big{(}\mathcal{H}_{\infty}^{J}(\omega)\big{)}$ of the
finite dimensional Banach space $\mathcal{H}_{\infty}^{J}(\omega)$ equipped
with the norm $\|D\|=\sup_{\text{Re}\,s>0}|D(s)|$. Based on harmonic analysis
on $\omega$-Dirichlet groups, we prove the formula
$\boldsymbol{\lambda}\big{(}\mathcal{H}_{\infty}^{J}(\omega)\big{)}~{}=~{}\displaystyle\lim_{T\to\infty}\frac{1}{2T}\int_{-T}^{T}\Big{|}\sum_{n\in
J}e^{-i\omega_{n}t}\Big{|}\,dt\,,$ and apply it to various concrete
frequencies $\omega$ and index sets $J$. To see an example, combining with a
recent deep result of Harper from probabilistic analytic number theory, we for
the space $\mathcal{H}_{\infty}^{\leq x}\big{(}(\log n)\big{)}$ of all
ordinary Dirichlet polynomials $D(s)=\sum_{n\leq x}a_{n}n^{-s}$ of length $x$
show the asymptotically correct order
$\boldsymbol{\lambda}\big{(}\mathcal{H}_{\infty}^{\leq x}\big{(}(\log
n)\big{)}\big{)}\sim\sqrt{x}/(\log\log x)^{\frac{1}{4}}.$

###### Key words and phrases:

Spaces of Dirichlet series, Projection constants

###### 2020 Mathematics Subject Classification:

Primary: 30B50, 43A77, 46B06. Secondary: 46B07; 46B07; 46G25.

The research of the fourth author was supported by the National Science Centre
(NCN), Poland, Project 2019/33/B/ST1/00165; the second, third and fifth author
were supported by CONICET-PIP 11220200102336 and PICT 2018-4250. The research
of the fifth author is additionally supported by ING-586-UNR

## Introduction

The study of complemented subspaces $X$ of a Banach space $Y$ and their
projection constants has a long history going back to the beginning of
operator theory in Banach spaces.

If $X$ is a complemented subspace of a Banach space $Y$, then the relative
projection constant of $X$ in $Y$ is defined by

$\boldsymbol{\lambda}(X,Y)=\inf\big{\\{}\|P\|:\,\,P\in
L(Y,X),\,\,P|_{X}=\operatorname{\mathrm{id}}_{X}\big{\\}}\,,$

where $\operatorname{\mathrm{id}}_{X}$ is the identity operator on $X$. The
following straightforward result shows the intimate link between projection
constants and extensions of linear operators: For every Banach space $Y$ and
its subspace $X$ one has

$\displaystyle\boldsymbol{\lambda}(X,Y)=\inf\big{\\{}c>0:\,\,\text{$\forall\,T\in\mathcal{L}(X,Z)$
\,\, $\exists$\,\, an extension\, $\widetilde{T}\in\mathcal{L}(Y,Z)$\, with
$\|\widetilde{T}\|\leq c\,\|T\|$}\big{\\}}\,.$

Moreover, the (absolute) projection constant of a Banach space $X$ is given by

$\boldsymbol{\lambda}(X):=\sup\,\,\boldsymbol{\lambda}(I(X),Y)\,,$

where the supremum is taken over all Banach spaces $Y$ and isometric
embeddings $I\colon X\to Y$.

A frequency $\omega=(\omega_{n})_{n\in\mathbb{N}}$ is a strictly increasing,
non-negative real sequence such that $\omega_{n}\to\infty$ as $n\to\infty$.
Given a finite index set $J\subset\mathbb{N}$ and complex numbers
$(a_{n})_{n\in J}$, we say that

$D(s):=\sum_{n\in J}a_{n}e^{-\omega_{n}s},\quad\,s\in\mathbb{C}$

is a $\omega$-Dirichlet polynomial supported on the index set $J$. For the
frequency $\omega=(n)_{n\in\mathbb{N}}$ we obtain (after the substitution
$z=e^{-s}$) polynomials $\sum_{n\in J}a_{n}z^{n}$ in one complex variable, and
in the case $\omega=(\log n)_{n\in\mathbb{N}}$ all ordinary Dirichlet
polynomials $\sum_{n\in J}a_{n}n^{-s}$.

Denote by $\mathcal{H}_{\infty}^{J}(\omega)$ the (finite dimensional) Banach
space of all $\omega$-Dirichlet polynomials supported on the finite index
subset $J\subset\mathbb{N}$, endowed with the norm

$\|D\|_{\infty}:=\sup_{t\in\mathbb{R}}\Big{|}\sum_{n\in
J}a_{n}e^{-i\omega_{n}t}\Big{|}=\sup_{\operatorname{Re}s>0}\Big{|}\sum_{n\in
J}a_{n}e^{-\omega_{n}s}\Big{|}\,,$

where the last equality is a simple consequence of the maximum modulus
principle.

Then the main goal of this article is to study the projection constant

$\boldsymbol{\lambda}\big{(}\mathcal{H}_{\infty}^{J}(\omega)\big{)}$

for various ’natural’ frequencies $\omega$ and various ’natural’ finite index
sets $J$ of $\mathbb{N}$. Given $x\in\mathbb{N}$, we are particularly
interested in the projection constant of the Banach space

$\mathcal{H}_{\infty}^{\leq
x}(\omega)=\mathcal{H}_{\infty}^{\\{n\in\mathbb{N}\colon n\leq
x\\}}(\omega)\,,$

so all $\omega$-Dirichlet polynomials $D(s)=\sum_{n\leq
x}a_{n}e^{-\omega_{n}s}$ of length $x$.

Before we illustrate some of our main results, let us pause for a moment to
say a few words about the modern theory of Dirichlet series. Within the last
two decades, the theory of ordinary Dirichlet series $\sum a_{n}n^{-s}$
experienced a kind of renaissance. The study of these series in fact was one
of the hot topics in mathematics at the beginning of the 20th century. Among
others, H. Bohr, Besicovitch, Bohnenblust, Hardy, Hille, Landau, Perron, and
M. Riesz.

However, this research took place before the modern interplay between function
theory and functional analysis, as well as the advent of the field of several
complex variables, and the area was in many ways dormant until the late 1990s.
One of the main goals of the 1997 paper of Hedenmalm, Lindqvist, and Seip [21]
was to initiate a systematic study of Dirichlet series from the point of view
of modern operator-related function theory and harmonic analysis.
Independently, at the same time, a paper of Boas and Khavinson [4] attracted
renewed attention, in the context of several complex variables, to the
original work of Bohr.

A new field emerged intertwining the classical work in novel ways with modern
functional analysis, infinite dimensional holomorphy, probability theory as
well as analytic number theory. As a consequence, a number of challenging
research problems crystallized and were solved over the last decades. We refer
to the monographs [9], [22], and [36], where many of the key elements of this
new developments for ordinary Dirichlet series are described in detail.

Contemporary research in this field owes much to the following fundamental
observation of H. Bohr [5], sometimes called Bohr’s vision:

By the transformation $z_{j}=\mathfrak{p}_{j}^{-s}$ and the fundamental
theorem of arithmetics, an ordinary Dirichlet series $\sum a_{n}n^{-s}$ may be
thought of as a function
$\sum_{\alpha\in\mathbb{N}_{0}^{(\mathbb{N})}}a_{\mathfrak{p}^{\alpha}}z^{\alpha}$
of infinitely many complex variables $z_{1},z_{2},\ldots$, where
$\mathfrak{p}=(\mathfrak{p}_{j})$ stands for the sequence of prime numbers. By
a classical approximation theorem of Kronecker, this is more than just a
formal transformation: If, say, only a finite number of the coefficients
$a_{n}$ are nonzero (so that questions about convergence of the series are
avoided), the supremum of the Dirichlet polynomial $\sum a_{n}n^{-s}$ in the
half-plane $\operatorname{Re}s>0$ equals the supremum of the corresponding
polynomial on the infinite-dimensional circle group $\mathbb{T}^{\infty}$.

Let us sketch some of our main results, as well as some ideas for the
strategies how to derive them.

Bohr’s vision in its original form is the seed which allows to associate with
each frequency $\omega$ a compact abelian group $G$ (a so-called
$\omega$-Dirichlet group) as well as a sequence of characters $h_{\omega_{n}}$
on $G$, such that for each finite set $J\subset\mathbb{N}$ the mapping
$\sum_{J}a_{n}e^{-\omega_{n}s}\mapsto\sum_{J}a_{n}h_{\omega_{n}}$ leads to a
coefficient preserving identification of $\mathcal{H}_{\infty}^{J}(\omega)$
with the Banach space $\text{Trig}_{J}(G)$ of all trigonometric polynomials on
$G$ having Fourier coefficients supported in $\\{h_{\omega_{n}}\colon n\in
J\\}$ (see Section 3.1).

Using a famous averaging technique of Rudin (Theorem 2.1 and Theorem 2.2), we
derive the following integral formula (see Theorem 3.6)

$\displaystyle\boldsymbol{\lambda}\big{(}\mathcal{H}_{\infty}^{J}(\omega)\big{)}=\int_{G}\Big{|}\sum_{n\in
J}h_{\omega_{n}}\Big{|}\,d\mathrm{m}\,,$

where $\mathrm{m}$ stands for the Haar measure on $G$, and then equivalently

$\displaystyle\boldsymbol{\lambda}\big{(}\mathcal{H}_{\infty}^{J}(\omega)\big{)}=\lim_{T\to\infty}\frac{1}{2T}\int_{-T}^{T}\Big{|}\sum_{n\in
J}e^{-i\omega_{n}t}\Big{|}\,dt$

(Theorem 3.1). Our main applications for concrete frequencies $\omega$ and
concrete index sets $J$ offer concrete estimates for
$\boldsymbol{\lambda}\big{(}\mathcal{H}_{\infty}^{J}(\omega)\big{)}$, and
their proofs are all mainly based on caculating one of the preceding two
integrals.

For the three standard frequencies $\omega=(n)_{n\in\mathbb{N}_{0}}$,
$\omega=(\log p_{n})_{n\in\mathbb{N}}$ (where as above $p_{n}$ is the $n$th
prime number), and $\omega=(\log n)_{n\in\mathbb{N}}$, we in Section 3.2, Case
I, get the following formulas which are asymptotically correct in $x$:

  * $\bullet$

$\boldsymbol{\lambda}\big{(}\mathcal{H}_{\infty}^{\leq
x}\big{(}(n)\big{)}\big{)}=\frac{4}{\pi^{2}}\log(x+1)+o(1)$ ,

  * $\bullet$

$\lim_{x\to\infty}\frac{\boldsymbol{\lambda}\big{(}\mathcal{H}_{\infty}^{\leq
x}\big{(}(\log p_{n})\big{)}\big{)}}{\sqrt{x}}=\frac{\sqrt{\pi}}{2}$ ,

  * $\bullet$

$\boldsymbol{\lambda}\big{(}\mathcal{H}_{\infty}^{\leq x}\big{(}(\log
n)\big{)}\big{)}=O\left(\frac{\sqrt{x}}{(\log\log x)^{\frac{1}{4}}}\right)$ .

To estimate the integrals for the first two statements is fairly standard –
but for the third one this is highly non-trivial. Indeed, a recent deep
theorem from probabilistic analytic number theory due to Harper from [19]
shows that

(1)
$\lim_{T\to\infty}\frac{1}{2T}\int_{-T}^{T}\Big{|}\sum_{n=1}^{x}\frac{1}{n^{it}}\Big{|}dt=O\left(\frac{\sqrt{x}}{(\log\log
x)^{\frac{1}{4}}}\right)\,,$

which is equivalent to

$\int_{\mathbb{T}^{\infty}}\Bigg{|}\sum_{\alpha\in\mathbb{N}^{(\mathbb{N})}_{0}:1\leq
n\leq x}z^{\alpha}\Bigg{|}\,dz=O\left(\frac{\sqrt{x}}{(\log\log
x)^{\frac{1}{4}}}\right)\,.$

This resolved a long-standing problem of Helson from [23]. In fact, Helson had
conjectured that the integral is of order $o(\sqrt{x})$ which would have
disproved a certain generalisation of Nehari’s theorem from harmonic analysis.
We also mention that Harper’s result gave a negative answer to the so-called
embedding problem showing that for $0~{}<p~{}<~{}2$ the $L_{p}$-integral of
every ordinary Dirichlet polynomial $D=\sum_{n\leq x}a_{n}n^{-s}$ over any
segment of fixed length on the vertical line $[\operatorname{Re}s=1/2]$ can
not be bounded by a universal constant times
$\lim_{T\to\infty}\frac{1}{2T}\int_{-T}^{T}|D(it)|^{p}dt$ (see also Problem
2.1 in [43]).

Finally, we mention that we also study the projection constant of Banach
spaces of ordinary Dirichlet polynomials supported on an index sets of natural
numbers with a certain complexity of their prime number decompositions (see
again Section 3.2, Case II and III).

We finish the article comparing and linking the results we obtained for
projection constants, with some important estimates of the unconditional basis
constant
$\boldsymbol{\chi_{\operatorname{mon}}}(\mathcal{H}_{\infty}^{J}(\omega))$
(see Section 4 for the definition), for several natural frequencies
$\omega=(\omega_{n})$ and index sets $J\subset\mathbb{N}$. From the group
point of view, this is related to the Sidon constant of specific sets of
characters.

##  1\. Preliminaries

We use standard notation from Banach space theory as e.g., used in the
monographs [11, 33, 35, 46, 48]. If not indicated differently, we consider
complex Banach spaces.

Given two sequences $(a_{n})$ and $(b_{n})$ of non-negative real numbers we
write $a_{n}\prec b_{n}$, if there is a constant $c>0$ such that $a_{n}\leq
c\,b_{n}$ for all $n\in\mathbb{N}$, while $a_{n}\sim b_{n}$ means that
$a_{n}\prec b_{n}$ and $b_{n}\prec a_{n}$ holds. In the case that an extra
parameter $m$ is also involved, for two sequences of non-negative real numbers
$(a_{n,m})$ and $(b_{n,m})$, we write $a_{n,m}\prec_{C(m)}b_{n,m}$ whenever
there is a constant $C(m)>0$ (which depends exclusively on $m$ and not on $n$)
such that $a_{n,m}\leq C(m)b_{n,m}$ for all $n,m\in\mathbb{N}$. We use the
notation $a_{n,m}\sim_{C(m)}b_{n,m}$ if $a_{n,m}\prec_{C(m)}b_{n,m}$ and
$b_{n,m}\succ_{C(m)}a_{n,m}$. We also write $a_{n,m}\prec_{C^{m}}b_{n,m}$ when
there is a hypercontractive comparision, i.e., there is an absolute constant
$C>0$ such that $a_{n,m}\leq C^{m}b_{n,m}$ for all $n,m\in\mathbb{N}$. Of
course, if $a_{n,m}\prec_{C^{m}}b_{n,m}$ and $b_{n,m}\succ_{C^{m}}a_{n,m}$ we
simply write $a_{n,m}\sim_{C^{m}}b_{n,m}$.

In the following two paragraphs we collect a few standard facts on projection
constants as well as topological groups.

###  1.1. Projection constants

Any Banach space $X$ can be embedded isometrically into $\ell_{\infty}(S)$,
where $S$ is a nonempty set which in general depends on $X$. For every
separable $X$, we may choose $S=\mathbb{N}$. Throughout the paper, we use the
fact that, if $S$ is a nonempty set for which the Banach space $X$ is
isometrically isomorphic to a subspace $Z$ of $\ell_{\infty}(S)$, then
$\boldsymbol{\lambda}(X)=\boldsymbol{\lambda}(Z,\ell_{\infty}(S))\,.$ Indeed,
this is due to the fact that $\ell_{\infty}(S)$ is isometrically injective
(see [1, Definition 2.5.1. and Proposition 2.5.2.]). Thus finding
$\boldsymbol{\lambda}(X)$ is equivalent to finding the norm of a minimal
projection from $\ell_{\infty}(S)$ onto an isometric copy of $X$ in
$\ell_{\infty}(S)$. However, this is a non-trivial problem in general.

General bounds for projection constants of various finite dimensional Banach
spaces were studied by many authors. The most fundamental general upper bound
is due to Kadets and Snobar [24]: For every $n$-dimensional Banach space
$X_{n}$ one has

(2) $\boldsymbol{\lambda}(X_{n})\leq\sqrt{n}\,.$

In contrast, König and Lewis [27] showed that for any Banach space $X_{n}$ of
dimension $n\geq 2$ the strict inequality
$\boldsymbol{\lambda}(X_{n})<\sqrt{n}$ holds, and this estimate was improved
by Lewis [30] showing

$\boldsymbol{\lambda}(X_{n})\leq\sqrt{n}\,\left(1-\frac{1}{n^{2}}\bigg{(}\frac{1}{5}\bigg{)}^{2n+11}\right)\,.$

The exact values of $\boldsymbol{\lambda}(\ell_{2}^{n})$ and
$\boldsymbol{\lambda}(\ell_{1}^{n})$ were computed by Grünbaum [18] and
Rutovitz [42]: In the complex case

(3)
$\displaystyle\boldsymbol{\lambda}\big{(}\ell_{2}^{n}(\mathbb{C})\big{)}=n\int_{\mathbb{S}_{n}(\mathbb{C})}|x_{1}|\,d\sigma=\frac{\sqrt{\pi}}{2}\frac{n!}{\Gamma(n+\frac{1}{2})}\,,$

where $d\sigma$ stands for the normalized surface measure on the sphere
$\mathbb{S}_{n}(\mathbb{C})$ in $\mathbb{C}^{n}$, and

(4)
$\displaystyle\boldsymbol{\lambda}\big{(}\ell_{1}^{n}(\mathbb{C})\big{)}=\int_{\mathbb{T}^{n}}\Big{|}\sum_{k=1}^{n}z_{k}\Big{|}\,dz=\int_{0}^{\infty}\frac{1-J_{0}(t)^{n}}{t^{2}}dt\,,$

where $dz$ denotes the normalized Lebesgue measure on the distinguished
boundary $\mathbb{T}^{n}$ in $\mathbb{C}^{n}$ and $J_{0}$ is the zero Bessel
function defined by
$J_{0}(t)=\frac{1}{2\pi}\int_{0}^{\infty}\cos(t\cos\varphi)d\varphi\,.$ The
corresponding real constants are different:

(5)
$\displaystyle\boldsymbol{\lambda}\big{(}\ell_{2}^{n}(\mathbb{R})\big{)}=n\int_{\mathbb{S}_{n}(\mathbb{R})}|x_{1}|\,d\sigma=\frac{2}{\sqrt{\pi}}\frac{\Gamma(\frac{n+2}{2})}{\Gamma(\frac{n+1}{2})}$
(6)
$\displaystyle\boldsymbol{\lambda}\big{(}\ell_{1}^{n}(\mathbb{R})\big{)}=\begin{cases}\boldsymbol{\lambda}\big{(}\ell_{2}^{n}(\mathbb{R})\big{)},&\text{$n$
odd}\\\\[5.69054pt]
\boldsymbol{\lambda}(\ell_{2}^{n-1}(\mathbb{R})),&\text{$n$
even}\,.\\\\[5.69054pt] \end{cases}$

Gordon [15] and Garling-Gordon [14] determined the asymptotic growth of
$\boldsymbol{\lambda}\big{(}\ell_{p}^{n}\big{)}$ for $1<p<\infty$ with
$p\notin\\{1,2,\infty\\}$:

(7) $\boldsymbol{\lambda}\big{(}\ell^{n}_{p}\big{)}\sim
n^{\min\big{\\{}\frac{1}{2},\frac{1}{p}\big{\\}}}\,.$

König, Schütt and Tomczak-Jagermann [28] proved that for $1\leq p\leq 2$

(8)
$\displaystyle\lim_{n\to\infty}\frac{\boldsymbol{\lambda}\big{(}\ell_{p}^{n}\big{)}}{\sqrt{n}}=\gamma\,,$

where $\gamma=\sqrt{\frac{2}{\pi}}$ in the real and
$\gamma=\frac{\sqrt{\pi}}{2}$ in the complex case. For an extensive treatment
on all of this and more see the excellent monograph [46] of Tomczak-
Jaegermann.

###  1.2. Topological groups

As usual a group $G$ equipped with a topology $\tau$ is said to be a
topological group whenever the mapping $(G,\tau)\times(G,\tau)\ni(x,y)\to
xy^{-1}\in(G,\tau)$ is continuous. From here on $G$ is assumed to be a compact
group, that is, its topology is compact. In this case, $G$ defines a natural
set of maps $\\{L_{g}\\}_{g\in G}$ and $\\{R_{g}\\}_{g\in G}$ on $C(G)$, the
complex-valued continuous functions on $G$, given for all $g,h\in G$ by

$L_{g}f(h):=f(gh),\quad\,\,\,\,\text{and\, $R_{g}f(h)=f(hg)$},\quad\,f\in
C(G)\,.$

It is well-known that for every compact group $G$ there exists a unique Borel
probability measure $\mathrm{m}$ which is left invariant, that is,

$\int_{G}f\,d\mathrm{m}=\int_{G}(L_{g}f)\,d\mathrm{m},\quad\,g\in G,\,\,\,f\in
C(G)\,.$

This $\mathrm{m}$ is called the Haar measure of $G$. If in addition
$\mathrm{m}$ is also right invariant:

$\int_{G}f\,d\mathrm{m}=\int_{G}(R_{g}f)\,d\mathrm{m},\quad\,g\in G,\,\,\,f\in
C(G)\,,$

then the compact group $G$ is called unimodular. Examples of unimodular groups
are compact groups in which every one point set is closed.

Let $\mathrm{m}$ be the normalized Haar measure on $G$, and $\widehat{G}$ as
usual the dual group of $G$ (i.e., the set of all continuous characters on
$G$). For any $f\in L^{1}(G):=L^{1}(G,\mathrm{m})$, the Fourier transform of
$f$ is given by

$\widehat{f}(\gamma):=\int_{G}\,f(x)\,\overline{\gamma(x)}\,d\mathrm{m}(x),\quad\,\gamma\in\widehat{G}\,.$

Recall that $L_{1}(G)$ forms a commutative Banach algebra, whenever it carries
the convolution $f_{1}\ast f_{2}$ as its multiplication, that is, for
$\mathrm{m}$-almost every $x\in G$

$(f_{1}\ast f_{2})(x):=\int_{G}f_{1}(xy^{-1})f_{2}(y)\,d\mathrm{m}(y)\,.$

Products of groups are going to be of particular interest for our purposes.
Given compact abelian groups $G_{1},\ldots,G_{n}$, each with the Haar measure
$\mathrm{m}_{j},1\leq j\leq n$, we denote by $G:=G_{1}\times\cdots\times
G_{n}$ the product of these groups endowed its natural product operation and
product topology . Given an $n$-tuple of characters
$(\gamma_{1},\ldots,\gamma_{n})\in\widehat{G_{1}}\times\cdots\times\widehat{G_{n}}$
and $\alpha=(\alpha_{1},\ldots,\alpha_{n})\in\mathbb{Z}^{n}$, we write
$\gamma^{\alpha}$ for the character in $\widehat{G}$ given by

$\gamma^{\alpha}(x_{1},\ldots,x_{n}):=\gamma_{1}(x_{1})^{\alpha_{1}}\cdots\gamma_{n}(x_{n})^{\alpha_{n}},\quad\,(x_{1},\ldots,x_{n})\in
G\,.$

Of special interest is the $n$-dimensional circle group $G:=\mathbb{T}^{n}$,
where the Haar measure $\mathrm{m}=:dz$ on $\mathbb{T}^{n}$ acts on a Borel
function $f\colon\mathbb{T}^{n}\to\mathbb{C}$ by the formula

$\int_{\mathbb{T}^{n}}f(z)\,dz=\frac{1}{(2\pi)^{n}}\int_{0}^{2\pi}\cdots\int_{0}^{2\pi}f(e^{it_{1}},\ldots,e^{it_{n}})\,dt_{1}\ldots
dt_{n}\,.$

Recall that $\widehat{\mathbb{T}^{n}}=\mathbb{Z}^{n}$, where the
identification is given by the fact that for every character
$\gamma\in\widehat{\mathbb{T}^{n}}$ there is a unique multi index
$(\alpha_{1},\ldots,\alpha_{n})\in\mathbb{Z}^{n}$ for which
$\gamma(z)=z^{\alpha}$ for every $z=(z_{1},\ldots,z_{n})\in\mathbb{T}^{n}$.

##  2\. Trigonometric polynomials

For any compact group $G$ and any nonempty finite set $E\subset\widehat{G}$ we
denote $\text{span}\,E$ by $\text{Trig}_{E}(G)\,,$ the space of all
trigonometric polynomials $P$ on $G$ which are supported on $E$, that is, the
Fourier transform $\hat{P}$ is supported on $E$. Together with the supremum
norm on $G$ this defines a (finite dimensional) Banach space. Note that each
such polynomial has the form

$P(g)\,=\,\sum_{\gamma\in E}\widehat{P}(\gamma)\,\gamma(g)\,,\quad g\in G\,.$

In the case of $E=\widehat{G}$, the space $\text{Trig}_{\widehat{G}}(G)$ of
all trigonometric polynomials on $G$ is denoted in short by $\text{Trig}(G)$.
We need two fundamental consequences of the Peter-Weyl theorem (see, e.g.,
[36, Theorem 1.3.3]): For any compact abelian group $G$, the space
$\text{Trig}(G)$ is dense in the space $C(G)$ of complex-valued functions (see
[36, Theorem 1.3.4]), and the dual group $\widehat{G}$ is an orthonormal basis
of the Hilbert space $L^{2}(G,\mathrm{m})$ (see [36, Theorem 1.3.6]).

For the special case $G=\mathbb{T}^{n}$ with $\widehat{G}=\mathbb{Z}^{n}$ we
are particularly interested in the index sets

(9)
$J_{p}(m,n):=\Big{\\{}\alpha\in\mathbb{Z}^{n}:\,\,\big{(}\sum_{j}|\alpha_{j}|^{p}\big{)}^{\frac{1}{p}}=m\Big{\\}},\quad\,\Lambda_{p}(m,n):=\Big{\\{}\alpha\in\mathbb{N}_{0}^{n}:\,\,\big{(}\sum_{j}\alpha_{j}^{p}\big{)}^{\frac{1}{p}}=m\Big{\\}}$

as well as

(10) $J_{p}(\leq
m,n):=\Big{\\{}\alpha\in\mathbb{Z}^{n}:\,\,\big{(}\sum_{j}\alpha_{j}^{p}\big{)}^{\frac{1}{p}}\leq
m\Big{\\}},\quad\,\Lambda_{p}(\leq
m,n):=\Big{\\{}\alpha\in\mathbb{N}_{0}^{n}:\,\,\big{(}\sum_{j}|\alpha_{j}|^{p}\big{)}^{\frac{1}{p}}\leq
m\Big{\\}}\,,$

where $m,n\in\mathbb{N}$ and $p\in\\{1,2,\infty\\}$. For $p=1$ we write
$|\alpha|=\sum_{j}|\alpha_{j}|$, usually called the order of $\alpha$, and for
$p=\infty$ we of course by $\big{(}\sum_{j}|\alpha_{j}|^{p}\big{)}^{1/p}$ mean
$\max_{j}|\alpha_{j}|$.

For $m~{}\in~{}\mathbb{N}$ we write
$\text{Trig}_{m}(\mathbb{T}^{n}):=\text{Trig}_{\Lambda_{1}(m,n)}(\mathbb{T}^{n})$
for the Banach space of all analytic trigonometric polynomials which are
$m$-homogeneous, i.e., all polynomials of the form

$P(z)=\sum_{\alpha\in\Lambda_{1}(m,n)}c_{\alpha}z^{\alpha}\quad\text{for}\,\,z\in\mathbb{T}^{n}\,.$

Similarly, $\text{Trig}_{\leq
m}(\mathbb{T}^{n}):=\text{Trig}_{\Lambda_{1}(\leq m,n)}(\mathbb{T}^{n})$,
stands for the space of all analytic trigonometric polynomials of degree $\leq
m$.

The following integral formula for the projection constant of the Banach space
of all trigonometric polynomials on a given compact abelian group supported on
a priori given set of characters in $\widehat{G}$, is one of the main sources
of what we intend to do.

######  Theorem 2.1.

Let $G$ be a compact abelian group and
$E:=\\{\gamma_{1},\ldots,\gamma_{N}\\}\subset\widehat{G}$ a finite set of
characters. Then $P\colon C(G)\to C(G)$, given by
$Pf=\sum_{j=1}^{N}\widehat{f}(\gamma_{j})\gamma_{j}$ for all $f\in C(G)$, is
the unique projection onto $\text{Trig}_{E}(G)$ that commutes with the action
of the group on $C(G)$. Moreover, $P$ is minimal:

$\boldsymbol{\lambda}\big{(}\text{Trig}_{E}(G)\big{)}=\big{\|}P\colon C(G)\to
C(G)\big{\|}=\int_{G}\Big{|}\sum_{j=1}^{N}\gamma_{j}(x)\Big{|}\,d\mathrm{m}(x)\,.$

The following section is devoted to the proof of this result.

###  2.1. Averaging projections

As mentioned, one of the main tools we intend to use is a method due to Rudin
(see the forthcoming Theorem 2.2). Roughly speaking, under certain
assumptions, there is a somewhat universal averaging technique which allows to
construct new projections with additional and somehow better properties from
an a priori given projection.

Given a topological group $G$ and a Banach space $Y$, we need to explain when
all elements of $G$ act as bounded linear operators on $Y$. Formally this
means that there is a mapping

$T\colon G\to\mathcal{L}(Y)\,,\,\,\,\,g\mapsto T_{g}$

such that

$T_{e}=I_{Y},\quad\,T_{gh}=T_{g}T_{h},\quad\,g,\,\,h\in G$

and all mappings

(11) $g\ni G\mapsto T_{g}(y)\in Y,\quad\,\,\,y\in Y$

are continuous. Then $G$ is said to act on $Y$ through $T$ (or simply, $G$
acts on $Y$). If in addition all operators $T_{g},g\in G$ are isometries, then
we say that $G$ acts isometrically on $Y$. We say that $S\in\mathcal{L}(Y)$
commutes with the action of $G$ on $Y$ through $T$ whenever $S$ commutes with
all $T_{h},\,h\in G$.

######  Theorem 2.2.

Let $Y$ be a Banach space, $X$ a complemented subspace of $Y$, and
$\mathbf{Q}\colon Y\to Y$ a projection onto $X$. Suppose that $G$ is a compact
group with Haar measure $m$, which acts on $Y$ through $T$ such that $X$ is
invariant under the action of $G$, that is,

(12) $T_{g}(X)\subset X,\quad\,g\in G\,.$

Then $\mathbf{P}\colon Y\to Y$ given by

(13)
$\mathbf{P}(y):=\int_{G}T_{g^{-1}}\mathbf{Q}T_{g}(y)\,d\mathrm{m}(g),\quad\,y\in
Y\,,$

is a projection onto $X$ which commutes with the action of $G$ on $Y$, i.e.,
$T_{g}\mathbf{P}=\mathbf{P}T_{g}$ for all $g\in G$, and satisfies

$\|\mathbf{P}\|\,\leq\,\|\mathbf{Q}\|\,\,\,\sup_{g\in G}\|T_{g}\|^{2}\,.$

Moreover, if there is a unique projection on $Y$ onto $X$ that commutes with
the action of $G$ on $Y$, and if $G$ acts isometrically on $Y$, then
$\mathbf{P}$ given in the formula (13) is minimal, i.e.,

$\boldsymbol{\lambda}(X,Y)=\|\mathbf{P}\|\,.$

This result (in one way or the other) found various applications in the
literature (see, e.g., [26, 28, 29, 37, 32, 40, 41]. In the case of the circle
group, it can be traced back to Faber’s article [13]. For the sake of
completeness we include a proof, which is inspired by [40] (and also [48,
Theorem III.B.13]).

###### Proof of Theorem 2.2.

Note first, that, by the Banach-Steinhaus theorem, $\sup_{g\in
G}\|T_{g}\|<\infty$. Then, given $y\in Y$, the mapping $g\ni G\mapsto
T_{g^{-1}}\mathbf{Q}T_{g}\in\mathcal{L}(Y)$ is bounded and by (11) measurable
(being almost everywhere separably valued and weakly measurable), and hence
Bochner integrable. Consequently, $\mathbf{P}$ defines an operator on $Y$.
Moreover, from (12) we deduce for all $y\in Y$ and for all $x\in X$ that

$T_{g^{-1}}\mathbf{Q}T_{g}(y)\in X,\quad\,T_{g^{-1}}\mathbf{Q}T_{g}(x)=x\,,$

implying that $\mathbf{P}$ is a projection from $Y$ onto $X$. The hypothesis
that $G$ acts on $Y$ (through $T$) yields for all $h\in G$

$T_{h^{-1}}\mathbf{P}T_{h}=\int_{G}T_{h^{-1}}T_{g^{-1}}\mathbf{Q}T_{g}T_{h}\,d\mathrm{m}(g)=\int_{G}T_{(gh)^{-1}}\mathbf{Q}T_{gh}\,d\mathrm{m}(g)=\mathbf{P}\,,$

so $\mathbf{P}$ commutes with the action of $G$ on $Y$. Since for all $y\in Y$

$\|\mathbf{P}y\|_{Y}\leq\int_{G}\|T_{g^{-1}}\|\,\|\mathbf{Q}\,\|\|T_{g}\|\,\|y\|_{Y}\,d\mathrm{m}(g)\leq\|\mathbf{Q}\|\,\sup_{g\in
G}\|T_{g}\|^{2}\,\|y\|_{Y}\,,$

the required estimate for $\|\mathbf{P}\|$ follows. If moreover there is a
unique projection on $Y$ onto $X$ that commutes with the action of $G$ on $Y$,
and if $G$ acts isometrically on $X$, then the projection $\mathbf{P}$ from
(13) does not depend on $\mathbf{Q}$ and
$\|\mathbf{P}\|\,\leq\,\|\mathbf{Q}\|$ for all projections $\mathbf{Q}$ on $Y$
onto $X$, i.e., $\|\mathbf{P}\|\leq\boldsymbol{\lambda}(X,Y)$. This completes
the proof. ∎

###### Proof of Theorem 2.1.

Note first that $G$ in a natural way acts on $C(G)$ (in the sense of Section
2.1), where the action is given by the mapping
$T:G\to\mathcal{L}(C(G)),\,g\mapsto T_{g}$ with

$T_{g}f(h):=f(gh),\quad\,f\in C(G),\,\,h\in G\,.$

We claim that $P\colon C(G)\to C(G)$ is the unique projection onto
$\text{Trig}_{E}(G)$ that commutes with all translation operators $T_{g}$,
$g\in G$. To see this, assume that $Q\colon C(G)\to C(G)$ onto
$\text{Trig}_{E}(G)$ is a projection that commutes with all translation
operators. Then for all $\gamma,\gamma^{\prime}\in\widehat{G}$ one has

$\widehat{T_{g}Q\gamma}(\gamma^{\prime})=\widehat{QT_{g}\gamma}(\gamma^{\prime})\,.$

It is easy to check that
$\widehat{T_{g}Q\gamma}(\gamma^{\prime})=\gamma^{\prime}(g)\,\widehat{Q\gamma}(\gamma^{\prime})$
and
$\widehat{QT_{g}\gamma}(\gamma^{\prime})=\gamma(g)\,\widehat{Q\gamma}(\gamma^{\prime})$.
In consequence, we get

$\gamma^{\prime}(g)\widehat{Q\gamma}(\gamma^{\prime})=\gamma(g)\widehat{Q\gamma}(\gamma^{\prime}),\quad\,g\in
G\,.$

This implies that, for all $\gamma,\gamma^{\prime}\in\widehat{G}$ with
$\gamma\neq\gamma^{\prime}$, we have $\widehat{Q\gamma}(\gamma^{\prime})=0$.
Combining with the Peter-Weyl theorem, which states that $\widehat{G}$ forms
an orthonormal basis in the Hilbert space $L^{2}(G)$, we conclude that

$Q\gamma=\sum_{\gamma^{\prime}\in\widehat{G}}\widehat{Q\gamma}(\gamma^{\prime})\,\gamma^{\prime},\quad\,\gamma\in\widehat{G}\,,$

and consequently, for every character $\gamma\in\widehat{G}$ there is a scalar
$c_{\gamma}$ such that $Q\gamma=c_{\gamma}\gamma$.

Since $Q$ is a projection onto $\text{Trig}(G)$, $c_{\gamma}=0$ for all
$\gamma\in\widehat{G}\setminus E$, and $c_{\gamma}=1$ for all $\gamma\in E$.
In consequence, $Q\gamma=\gamma$ for all $\gamma\in E$, and $Q\gamma=0$ for
all $\gamma\in\widehat{G}\setminus E$. Hence the projection $Q$, restricted to
the algebra $\text{Trig}(G)$ of all trigonometric polynomials on $G$, has the
representation

(14) $\displaystyle
Qf=\sum_{j=1}^{N}\widehat{f}(\gamma_{j})\gamma_{j},\quad\,f\in\text{Trig}(G)\,.$

Consequently, we conclude from the density of $\text{Trig}(G)$ in $C(G)$ that
the above formula holds for all $f\in C(G)$. Hence $Q=P$. This proves the
claim. Since $P$ is the unique projection onto $\text{Trig}_{E}(G)$ that
commutes with all translation operators, $P$ is fixed by the averaging
technique introduced in Theorem 2.2. Now observe that $G$ acts isometrically
on $C(G)$, i.e., all mappings $T_{g}\colon C(G)\to C(G)$ are isometries on
$C(G)$. Since for all $g\in G$

$T_{g}\gamma=\gamma(g)\gamma,\quad\,\gamma\in\widehat{G}\,,$

it follows that $T_{g}(\text{Trig}_{E}(G))\subset\text{Trig}_{E}(G)$ for all
$g\in G$. Then, by the moreover part of Theorem 2.2, $P$ is a minimal
projection, that is,

$\boldsymbol{\lambda}\big{(}\text{Trig}_{E}(G)\big{)}=\big{\|}P\colon C(G)\to
C(G)\big{\|}\,.$

Finally, it remains to prove the integral formula for the norm of $P$. Since
$f\ast\gamma=\widehat{f}(\gamma)\gamma$ for all $f\in L^{1}(G)$ and
$\gamma\in\widehat{G}$, we get by (14),

$Pf=f\ast\Big{(}\sum_{j=1}^{N}\gamma_{j}\Big{)},\quad\,f\in C(G)\,.$

Clearly, $\sum_{j=1}^{N}\gamma_{j}\in C(G)$, so it can readily be shown by
direct computation that

$\big{\|}P\colon C(G)\to
C(G)\big{\|}=\int_{G}\,\Big{|}\sum_{j=1}^{N}\gamma_{j}(x)\Big{|}\,d\mathrm{m}(x)\,,$

and this completes the proof. ∎

###  2.2. Projection constants - trigonometric polynomials

Based on the integral formula from Theorem 2.1, we derive concrete formulas
and asymptotically optimal estimates for the projection constants of spaces of
trigonometric polynomials on compact abelian groups, which have Fourier
expansions supported on an a priori fixed set of characters.

####  2.2.1. $\Lambda(2)$-sets

In order to show a very first application of Theorem 2.1, we need some further
notation and preliminaries. Recall that Rudin in his classical paper [39] from
1960 introduced the notion of $\Lambda(p)$-sets within the setting of Fourier
analysis on the circle group $\mathbb{T}$. In modern language, if $G$ is a
compact abelian group (with Haar measure $\mathrm{m}$) and $p\in(1,\infty)$,
then the subset ${E}\subset\widehat{G}$ is said to be a $\Lambda(p)$-set
whenever there exists a constant $C>0$ such that, for every trigonometric
polynomial $P\in\text{Trig}_{E}(G)$, one has

(15) $\|P\|_{L_{p}(G)}\leq C\|P\|_{L_{1}(G)}\,.$

In this case, the least such constant is called the $\Lambda(p)$-constant of
$E$, and denoted by $C_{p}=C_{p}(E)$. Let us here remark that for $p>2$ the
validity of (15) is equivalent to the existence of a constant $A_{p}>0$ such
that

$\|P\|_{L_{p}(G)}\leq A_{p}\|P\|_{L_{2}(G)},\quad\,P\in\text{Trig}_{E}(G)\,.$

The following almost immediate consequence of Theorem 2.1 shows that
$\Lambda(2)$-sets are of particular importance for our purposes – see also
Corollary 2.5 and Corollary 3.9.

######  Corollary 2.3.

Let $G$ be a compact abelian group. Then, for any finite set
$E=\\{\gamma_{1},\ldots,\gamma_{N}\\}\subset\widehat{G}$ of different
characters with $\Lambda(2)$ constant $C_{2}$, we have

$C_{2}^{-1}\sqrt{N}\leq\boldsymbol{\lambda}\big{(}\text{Trig}_{E}(G)\big{)}\leq\sqrt{N}\,.$

###### Proof.

Let $\mathrm{m}$ be the Haar measure on $G$. Then from Theorem 2.1, we get

$\boldsymbol{\lambda}\big{(}\text{Trig}_{E}(G)\big{)}=\int_{G}|\gamma_{1}(x)+\ldots+\gamma_{N}(x)|\,d\mathrm{m}(x)\,.$

Since $\widehat{G}$ is an orthonormal basis in $L_{2}(G,\mathrm{m})$,

$\bigg{(}\int_{G}|\gamma_{1}(x)+\ldots+\gamma_{N}(x)|^{2}\,d\mathrm{m}(x)\bigg{)}^{\frac{1}{2}}=\sqrt{N}\,,$

and this completes the proof. ∎

In combination with the preceding corollary we need the following example.

######  Remark 2.4.

Let $G$ be a compact abelian group. Following [17], a set
$E\subset\widehat{G}$ is said to be a $B_{2}$-set, whenever
$\gamma_{1}\gamma_{2}=\gamma_{3}\gamma_{4}$ for all
$\gamma_{1},\ldots,\gamma_{4}\in E$ if and only if
$\\{\gamma_{3},\gamma_{4}\\}$ is a permutation of
$\\{\gamma_{1},\gamma_{2}\\}$. It is worth noting that $B_{2}$-sets are
$\Lambda(2)$-sets with $\Lambda(2)$-constant $\sqrt{2}$ (see [17, Proposition
6.3.11]).

In passing we observe that [17, Proposition 6.3.11], also implies the
following: If $E\subset\widehat{G}$ fulfills that there exists $N\geq 2$ such
that for every $\gamma\in\widehat{G}$ there are at most $N$ pairs of the form
$(\gamma_{i_{1}},\gamma_{i_{2}})\subset E\times E$ with
$\gamma_{i_{1}}\gamma_{i_{2}}=\gamma$, then $E$ is $\Lambda(2)$-set with
$C_{2}\leq\sqrt{N}$.

Another example of $\Lambda(2)$-sets, important for our purposes, follows from
an inequality due to Weissler [47] (see also [9, Theorem 8.10]): For all
$P\in\text{Trig}_{\leq m}(\mathbb{T}^{n})$

(16)
$\displaystyle\frac{1}{\sqrt{2}^{m}}\Big{(}\int_{\mathbb{T}^{n}}|P(z)|^{2}\,dz\Big{)}^{\frac{1}{2}}\leq\int_{\mathbb{T}^{n}}|P(z)|\,dz\,.$

In other terms, the set $\big{\\{}z^{\alpha}\colon\alpha\in\Lambda_{1}(\leq
m,n)\big{\\}}$ of characters in $\widehat{\mathbb{T}^{n}}=\mathbb{Z}^{n}$
forms a $\Lambda(2)$-set with constant $C_{2}\leq\sqrt{2^{m}}$ (recall from
(10) the definition of $\Lambda_{1}(\leq m,n)$, which clearly may be
identified with the set of characters $z^{\alpha}$ it generates).

Then the following result is an immediate consequence of Theorem 2.1,
Corollary 2.3, and (16).

######  Corollary 2.5.

Let $J\subset\mathbb{Z}_{0}^{n}$ be a finite set. Then

$\boldsymbol{\lambda}\big{(}\text{Trig}_{J}(\mathbb{T}^{n})\big{)}=\int_{\mathbb{T}^{n}}\Big{|}\sum_{\alpha\in
J}z^{\alpha}\Big{|}\,dz\,,$

and if $\Lambda\subset\Lambda_{1}(\leq m,n)$, then

$\frac{1}{\sqrt{2^{m}}}\sqrt{|\Lambda|}\leq\boldsymbol{\lambda}\big{(}\text{Trig}_{\Lambda}(\mathbb{T}^{n})\big{)}\leq\sqrt{|\Lambda|}\,.$

In the context of this corollary the following formula and estimate

(17) $|\Lambda_{1}(m,n)|=\dbinom{n+m-1}{m}\leq
e^{m}\Big{(}1+\frac{n}{m}\Big{)}^{m}$

for the cardinality of $\Lambda_{1}(m,n)$ is very useful.

####  2.2.2. Products of groups

The aim is to prove various multivariate variants of a famous result by
Lozinski-Kharshiladze (see [48, IIIB. Theorem 22] and [34]), which shows a
precise estimate of the projection constant of $\text{Trig}_{\leq
m}(\mathbb{T})$, the space of trigonometric polynomials on $\mathbb{T}$ of
degree at most $m$ equipped with the sup-norm:

(18) $\boldsymbol{\lambda}\big{(}\text{Trig}_{\leq
m}(\mathbb{T})\big{)}=\frac{4}{\pi^{2}}\log(m+1)+o(1).$

This is done by mostly routine applications of Theorem 2.1. We point out that
these results in Section 3.2 are the key to obtain formulas and estimates for
the projection constants of spaces of Dirichlet polynomials.

######  Proposition 2.6.

Let $G:=G_{1}\times\cdots\times G_{n}$ be a product of compact abelian groups
$G_{1},\ldots,G_{n}$ $($with the Haar measures $\mathrm{m}_{j},1\leq j\leq
n)$, and $E:=\big{\\{}\gamma^{\alpha}\colon\alpha\in J_{\infty}(\leq
m,n)\big{\\}}\subset\widehat{G}$, where
$(\gamma_{1},\ldots,\gamma_{n})\in\widehat{G_{1}}\times\cdots\times\widehat{G_{n}}$
and the definition of $J_{\infty}(\leq m,n)$ is given in (10). Then

$\boldsymbol{\lambda}\big{(}\text{Trig}_{{E}}(G)\big{)}=\prod_{j=1}^{n}\int_{G_{j}}\Big{|}\sum_{k\in\mathbb{Z}:|k|\leq
m}\gamma_{j}(x_{j})^{k}\Big{|}\,d{\mathrm{m}}_{j}(x_{j})\,.$

###### Proof.

Let $\mu$ be the Haar measure on $G$, which is nothing else then the product
of the Haar measures $\mathrm{m}_{j}$. Then Theorem 2.1 yields

$\boldsymbol{\lambda}\big{(}\text{Trig}_{{E}}(G)\big{)}=\int_{G}\Big{|}\sum_{\alpha\in
J_{\infty}(\leq m,n)}\gamma^{\alpha}(x)\Big{|}\,d\mu(x)\,.$

Clearly, for all $x=(x_{1},\ldots,x_{n})\in G_{1}\times\cdots\times G_{n}$ one
has

$\sum_{\alpha\in J_{\infty}(\leq
m,n)}\gamma^{\alpha}(x)=\prod_{j=1}^{n}\sum_{|k|\leq
m}\gamma_{j}(x_{j})^{k}\,,$

and hence

$\boldsymbol{\lambda}\big{(}\text{Trig}_{{E}}(G)\big{)}=\int_{G}\prod_{j=1}^{n}\Big{|}\sum_{|k|\leq
m}\gamma_{j}(x_{j})^{k}\Big{|}\,d\mu(x)\,.$

Fubini’s theorem finishes the argument. ∎

We also use the following more general variant.

######  Proposition 2.7.

Let $G:=G_{1}\times\cdots\times G_{n}$ be a product of compact abelian groups
$G_{1},\ldots,G_{n}$. Moreover. let $I:=I_{1}\times\dots\times
I_{n}\subset\mathbb{Z}^{n}$, with $I_{j}$ finite subsets of $\mathbb{Z}$ for
$j\in\\{1,\ldots,n\\}$, and $E:=\\{\gamma^{\alpha}\colon\alpha\in I\\}$. Then

$\boldsymbol{\lambda}\big{(}\text{Trig}_{{E}}(G)\big{)}=\prod_{j=1}^{n}\int_{G_{j}}\Big{|}\sum_{k\in
I_{j}}\gamma_{j}(x_{j})^{k}\Big{|}\,d{\mathrm{m}}_{j}(x_{j})\,.$

As announced above, we finally present multivariate variants of the the
Lozinski-Kharshiladze result mentioned in (18). For each $m\in\mathbb{N}_{0}$,
let $D_{m}$ be the $m$th Dirichlet kernel

$D_{m}(t):=\sum_{k=-m}^{m}e^{-ikt}\,,$

and $L_{m}$ the $m$th Lebesgue constant given by

$L_{m}:=\frac{1}{2\pi}\int_{0}^{2\pi}|D_{m}(t)|\,dt=\frac{1}{2\pi}\int_{0}^{2\pi}\bigg{|}\frac{\sin(m+\frac{1}{2})t}{\sin\frac{t}{2}}\bigg{|}\,dt\,.$

We recall the well-known standard estimates

(19) $\frac{4}{\pi^{2}}\log(m+1)\,\,<\,\,L_{m}\,\,<\,\,3+\log
m,\quad\,m\in\mathbb{N}\,.$

######  Corollary 2.8.

For $\mathbf{d}=(d_{1},\dots,d_{n})\in\mathbb{N}_{0}^{n}$ let
$I_{\mathbf{d}}:=\big{\\{}\alpha\in\mathbb{Z}^{n}\colon\,|\alpha_{j}|\leq
d_{j},\,\,1\leq j\leq n\big{\\}}$. Then

$\boldsymbol{\lambda}\big{(}\text{Trig}_{I_{\mathbf{d}}}(\mathbb{T}^{n})\big{)}=\prod_{j=1}^{n}L_{d_{j}}\,.$

###### Proof.

Applying Proposition 2.7, we get

$\displaystyle\boldsymbol{\lambda}\big{(}\text{Trig}_{I_{\mathbf{d}}}(\mathbb{T}^{n})\big{)}=\prod_{j=1}^{n}\int_{\mathbb{T}}\Big{|}\sum_{|\alpha_{j}|\leq
d_{j}}z^{\alpha_{j}}\Big{|}\,dz=\prod_{j=1}^{n}\int_{\mathbb{T}}|D_{d_{j}}(z_{j})|\,dz_{j}=\prod_{j=1}^{n}L_{d_{j}}\,,$

as required. ∎

In order to state the ’analytic counterpart’ of Corollary 2.8, we define for
each $m\in\mathbb{N}_{0}$,

$D_{m}^{+}(t):=\sum_{k=0}^{m}e^{-ikt}$ and
$L_{m}^{+}:=\frac{1}{2\pi}\int_{0}^{2\pi}|D_{m}^{+}(t)|\,dt$ .

and recall from (10) the notation $\Lambda_{\infty}(\leq
m,n):=\big{\\{}\alpha\in\mathbb{N}_{0}^{n}:\,\,\max_{j}\alpha_{j}\leq
m\big{\\}}$.

######  Corollary 2.9.

For $m,n\in\mathbb{N}$

  * (i)

$\boldsymbol{\lambda}\big{(}\text{Trig}_{\Lambda_{\infty}(\leq
m,n)}(\mathbb{T}^{n})\big{)}=(L_{m}^{+})^{n}$ ,

  * (ii)

$\boldsymbol{\lambda}\big{(}\text{Trig}_{\Lambda_{\infty}(\leq
2m,n)}(\mathbb{T}^{n})\big{)}=(L_{m})^{n}$ ,

  * (iii)

$(L_{m}-1)^{n}\leq\boldsymbol{\lambda}\big{(}\text{Trig}_{\Lambda_{\infty}(\leq
2m+1),n)}(\mathbb{T}^{n})\big{)}\leq(L_{m}+1)^{n}$ ,

  * (iv)

$\boldsymbol{\lambda}\big{(}\text{Trig}_{\Lambda_{\infty}(\leq
m,n)}(\mathbb{T}^{n})\big{)}\sim(1+\log m)^{n}$ .

###### Proof.

(i) By Proposition 2.6, we have as desired

$\displaystyle\boldsymbol{\lambda}\big{(}\text{Trig}_{\Lambda_{\infty}(\leq
m,n)}(\mathbb{T}^{n})\big{)}=\prod_{j=1}^{n}\int_{\mathbb{T}}\Big{|}\sum_{0\leq\alpha_{j}\leq
m}z^{\alpha_{j}}\Big{|}\,dz=\prod_{j=1}^{n}\int_{\mathbb{T}}|D_{m}^{+}(z_{j})|\,dz_{j}=(L_{m}^{+})^{n}\,.$

(ii) It is easy to check that

$D_{2m}^{+}(t)=e^{-imt}\,\frac{\sin(m+\frac{1}{2})t}{\sin\frac{t}{2}},\quad\,t\in(0,2\pi)\,.$

This implies that
$L_{m}=\|D_{m}\|_{L_{1}(\mathbb{T})}=\|D_{2m}^{+}\|_{L_{1}(\mathbb{T})}=L_{2m}^{+}$,
so the statement follows from (i).

(iii) The statement follows by (ii) combined with the obvious estimates

$\|D_{2m}^{+}\|_{L_{1}(\mathbb{T})}-1\leq\|D_{2m+1}^{+}\|_{L_{1}(\mathbb{T})}\leq\|D_{2m}^{+}\|_{L_{1}(\mathbb{T})}+1,\quad\,m\in\mathbb{N}\,.$

(iv) Combining the estimates from (19) with those from (ii) and (iii), we get
the required equivalence. ∎

We conclude with the following two limit formulas; see again (10) for the
definition of the index sets $J_{\infty}(\leq m,n)$ and $\Lambda_{\infty}(\leq
m,n)$.

######  Corollary 2.10.

For each $m,n\in\mathbb{N}$

$\lim_{m\to\infty}\frac{\boldsymbol{\lambda}\big{(}\text{Trig}_{J_{\infty}(\leq
m,n)}(\mathbb{T}^{n})\big{)}}{\log^{n}m}=\lim_{m\to\infty}\frac{\boldsymbol{\lambda}\big{(}\text{Trig}_{\Lambda_{\infty}(\leq
m,n)}(\mathbb{T}^{n})\big{)}}{\log^{n}m}=\Big{(}\frac{4}{\pi^{2}}\Big{)}^{n}\,.$

###### Proof.

Recall the well-known formula

(20) $\lim_{m\to\infty}\frac{L_{m}}{\log m}=\frac{4}{\pi^{2}}\,.$

Then the first limit follows from Corollary 2.8. For the proof of the second
limit we distinguish the two sequences
$\big{(}\boldsymbol{\lambda}\big{(}\text{Trig}_{\Lambda_{\infty}(\leq
2m,n)}(\mathbb{T}^{n})\big{)}/\log(2m)\big{)}_{m}$ and
$\big{(}\boldsymbol{\lambda}\big{(}\text{Trig}_{\Lambda_{\infty}(\leq
2m+1,n)}(\mathbb{T}^{n})\big{)}/\log(2m+1)\big{)}_{m}$. Observe that both
sequences converge to $\frac{4}{\pi^{2}}$. Indeed, by Corollary 2.9 (ii) we
have that

$\boldsymbol{\lambda}\big{(}\text{Trig}_{\Lambda_{\infty}(\leq
2m,n)}(\mathbb{T}^{n})\big{)}=(L_{m})^{n}\,.$

So, using (20), gives the claim for the first sequence. The second sequence is
handled the same way using Corollary 2.9 (iii). ∎

####  2.2.3. Lebesgue constants

We point out that although we in Corollary 2.5 prove an integral formula for
the projection constant of trigonometric polynomials on $\mathbb{T}^{n}$
(supported on the index set $J$), we usually will not be able to compute that
integral explicitly.

In fact the integral from Corollary 2.5 appears naturally in multivariate
Fourier analysis (due to different summation methods of multivariate Fourier
series), and in this context it is usually called Lebesgue constant (of the
index set $J$).

Let us illustrate this with an example of particular interest. Given
$m,n\in\mathbb{N}$, the index set

(21) $J_{2}(\leq
m,n)=\Big{\\{}\alpha\in\mathbb{Z}^{n}:\,\sum_{j=1}^{n}|\alpha_{j}|^{2}\leq
m^{2}\Big{\\}}\,,$

of all spherical multi indices $\alpha\in\mathbb{Z}^{n}$ of ’Euclidean order’
less or equal $m$ (see again (10)) plays an outstanding role when multivariate
Fourier series are summed up by their spherical partial sums. Obviously, we
have that

$\boldsymbol{\lambda}\big{(}\text{Trig}_{J_{2}(\leq
m,n)}(\mathbb{T}^{n})\big{)}\leq\sqrt{\text{dim}\big{(}\text{Trig}_{J_{2}(\leq
m,n)}(\mathbb{T}^{n})\big{)}}=\sqrt{|J_{2}(\leq m,n)|}\,;$

this can either be seen as an immediate consequence of Corollary 2.5 or the
Kadec-Snobar theorem.

However, to find a precise formula for the dimension of
$\text{Trig}_{J_{2}(\leq m,n)}(\mathbb{T}^{n})$, or equivalently the
cardinality of $J_{2}(\leq m,n)$, is in general a highly non-trivial problem.

The so-called $n$-dimensional sphere problem of classical lattice point theory
concerns the number $\mathcal{N}_{n}(R)$ of lattice points in the closed
Euclidean ball of radius $R$ in $\mathbb{R}^{n}$. Let us mention here that in
the case of $n>3$ one has

$\mathcal{N}_{n}(R)=\omega_{n}R^{n}+O\big{(}R^{n-2}\big{)}\,,$

where $\omega_{n}:=\frac{2\pi^{n/2}}{\Gamma(n/2)}$ is the volume of the unit
ball in $\mathbb{R}^{n}$. For $n=3$ Heath-Brown proves in [20] that

$\mathcal{N}_{3}(R)=\frac{4}{3}\pi
R^{3}+O\big{(}R^{\frac{21}{16}+\varepsilon}\big{)}\,,$

where $\varepsilon>0$ is arbitrary. His proof is based on an explicit formula
for the number of ways to write $n\in\mathbb{N}$ as a sum of three squares of
integers. To get this, a Dirichlet $L$-series
$L(1,\chi_{n})=\sum_{m=1}^{\infty}\chi_{n}m^{-s}$ with
$\chi_{n}(m)=\big{(}-\frac{4n}{m}\big{)}$ is studied. We also refer to the
remarkable article [16] by Götze, where the lattice point problem is studied
for the more complicated case of $n$-dimensional ellipsoids.

Under the above notation of (21) we are ready to state the following
corollary.

######  Corollary 2.11.

There exist constants $C_{1}$ and $C_{2}$ with $C_{1}<C_{2}$ depending on $n$
such that

$C_{1}\,m^{\frac{n-1}{2}}\leq\,\boldsymbol{\lambda}\big{(}\text{Trig}_{J_{2}(\leq
m,n)}(\mathbb{T}^{n})\big{)}\leq\,C_{2}\,m^{\frac{n-1}{2}}\,.$

###### Proof.

We use a two-sided estimate proved by Babenko [2], which (in our notation)
states there exist constants $C_{1}$ and $C_{2}$ with $C_{1}<C_{2}$ depending
on $n$, such that

$C_{1}\,m^{\frac{n-1}{2}}\leq\,\int_{\mathbb{T}^{n}}\Big{|}\sum_{\alpha\in
J_{2}(\leq m,n)}z^{\alpha}\Big{|}\,dz\,\leq\,C_{2}\,m^{\frac{n-1}{2}}\,.$

Thus the required statement follows from Corollary 2.5. ∎

For the history of Babenko’s important result, we refer to the interesting
survey paper by Liflyand [31, Theorem 1.1]. We note that the proof given
there, does not give precise asymptotic estimates in the dimension $n$ for the
constants $C_{1}$ and $C_{2}$. The estimates of both $C_{1}$ and $C_{2}$
depend deeply on the Fourier transform of the indicator function of the closed
Euclidean ball of $\mathbb{R}^{n}$, which is expressed by the well-known
Bessel function. It seems interesting to note that
$C_{1}=C_{1}(\varepsilon):=C_{4}\varepsilon^{(n-1)/2}-C_{3}\varepsilon^{n/2}$
for small $\varepsilon\in(0,1)$.

##  3\. Dirichlet polynomials

Recall from the introduction that, given a frequency $\omega=(\omega_{n})$ and
a finite index set $J\subset\mathbb{N}$, we write
$\mathcal{H}_{\infty}^{J}(\omega)$ for the Banach space of all
$\omega$-Dirichlet polynomials supported on $J$ endowed with the supremum norm
on the imaginary line. The following integral formula for the projection
constant of $\mathcal{H}_{\infty}^{J}(\omega)$ is one of our main
contributions, and crucial for all further descriptions of
$\boldsymbol{\lambda}\big{(}\mathcal{H}_{\infty}^{J}(\omega)\big{)}$ for
concrete frequencies and index sets. See also Theorem 3.6 for an equivalent
formulation in terms of compact abelian groups.

######  Theorem 3.1.

Let $J\subset\mathbb{N}$ be a finite subset and $\omega=(\omega_{n})$ a
frequency. Then

$\displaystyle\boldsymbol{\lambda}\big{(}\mathcal{H}_{\infty}^{J}(\omega)\big{)}=\lim_{T\to\infty}\frac{1}{2T}\int_{-T}^{T}\Big{|}\sum_{n\in
J}e^{-i\omega_{n}t}\Big{|}\,dt\,.$

The proof, given at the end of the following section, is mainly based on
Theorem 2.1 and a reformulation of $\mathcal{H}_{\infty}^{J}(\omega)$ in terms
of a Hardy type space of trigonometric polynomials on certain compact abelian
groups associated to $\omega$, which are all supported on a finite set of
characters associated to $J$.

###  3.1. Bohr’s vision

We need to adapt ideas from [10]. Our first aim is to describe the finite
dimensional Banach space $\mathcal{H}_{\infty}^{J}(\omega)$ in terms of a
Hardy type space of functions on a certain compact abelian group.

In what follows, a pair $(G,\beta)$ is said to be a Dirichlet group if $G$ is
a compact abelian group and $\beta\colon\mathbb{R}\to G$ a continuous
homomorphism with dense range; as before we denote the Haar measure on $G$ by
$\mathrm{m}$. Then, by density, the dual map

$\widehat{\beta}\colon\widehat{G}\to\widehat{\mathbb{R}}$

is injective. Moreover, if we write $\mathbb{R}$ for the group
$(\mathbb{R},+)$ endowed with its natural topology, then the homomorphism
$\mathbb{R}=\widehat{\mathbb{R}}\,,\,\,\,x\mapsto e^{ix\,\bullet}$ identifies
$\mathbb{R}$ and $\widehat{\mathbb{R}}$ as topological groups, and hence
$\widehat{\beta}(\widehat{G})$ may be interpreted as a subset of $\mathbb{R}$.

We in particular observe that if $(G,\beta)$ is a Dirichlet group and
$\gamma\colon G\to\mathbb{T}$ a character, then the composition
$\gamma\circ\beta\colon\mathbb{R}\to\mathbb{T}$ is a character on
$\mathbb{R}$. Thus there exists a unique $x\in\mathbb{R}$ such that
$\gamma\circ\beta(t)=e^{ixt}$ for all $t\in\mathbb{R}$. The following result
from [10, Proposition 3.10] is crucial - we include a proof for the sake of
completeness.

######  Proposition 3.2.

For any Dirichlet group $(G,\beta)$ one has

$\int_{G}f(g)\,d\mathrm{m}(g)=\lim_{T\to\infty}\,\frac{1}{2T}\int_{-T}^{T}f\circ\beta(t)\,dt,\quad\,f\in
C(G)\,.$

###### Proof.

Note first that for all $\gamma\in\widehat{G}$ we have

$\int_{G}\gamma(g)\,d\mathrm{m}(g)=\lim_{T\to\infty}\,\frac{1}{2T}\int_{-T}^{T}\gamma\circ\beta(t)\,dt\,.$

Indeed, as explained above by the injectivity of the dual map
$\widehat{\beta}$, we know that for every $\gamma\in\widehat{G}$ there is a
unique $x=x(\gamma)\in\mathbb{R}$ such that $\gamma\circ\beta(t)=e^{ixt}$ for
all $t\in\mathbb{R}$. Now recall that $\int_{G}\gamma\,d\mathrm{m}=0$ (resp.,
$\int_{G}\gamma\,d\mathrm{m}=1$) whenever $\gamma\neq 1$ (resp., $\gamma=1$).
Clearly, $\gamma\neq 1$ (resp., $\gamma=1$) yields $x\neq 0$ (resp., $x=0$),
so the above equality is obvious in this case. As a consequence, the claim
also holds for all trigonometric polynomials $f\in C(G)$. But since all
trigonometric polynomials are dense in $C(G)$, and moreover the set
$\\{\Phi_{T}\colon T>0\\}$ of all linear functionals

$\Phi_{T}\colon
C(G)\to\mathbb{C}\,,\quad\,\Phi_{T}(f):=\frac{1}{2T}\int_{-T}^{T}f\circ\beta(t)dt\,$

is uniformly bounded, the conclusion is a consequence of the Banach-Steinhaus
theorem. ∎

Given a frequency $\omega=(\omega_{n})$, we need Dirichlet groups which are
adapted to $\omega$. We call the pair $(G,\beta)$ a $\omega$-Dirichlet group,
whenever $\\{\omega_{n}\colon
n\in\mathbb{N}\\}\subset\widehat{\beta}(\widehat{G})$, that is, for every
character $e^{-i\omega_{n}\bullet}:\mathbb{R}\to\mathbb{T}$ there is a
character $h_{\omega_{n}}\in\widehat{G}$ (which then is unique) such that the
following diagram commutes:

$G$$\mathbb{T}$$\mathbb{R}$$h_{\omega_{n}}$$e^{-i\omega_{n}\bullet}$$\beta$

For all needed information on $\omega$-Dirichlet groups see again [10].

Let us collect some examples. The very first is the frequency $\omega=(n)$ for
which the pair $(\mathbb{T},\beta_{\mathbb{T}})$ with

$\beta_{\mathbb{T}}:\mathbb{R}\rightarrow\mathbb{T}\,,\,\,t\mapsto e^{-it}$

obviously forms a $\omega$-Dirichlet group. Identifying
$\widehat{\mathbb{T}}=\mathbb{Z}$ we get that $h_{n}(z)=z^{n}$ for
$z\in\mathbb{T},n\in\mathbb{Z}$.

The second example shows that $\omega$-Dirichlet groups for any possible
frequency $\omega$ always exist. In fact, given a frequency $\omega$, the so-
called Bohr compactification of the reals forms a $\omega$-Dirichlet group.

######  Example 3.3.

Denote by $\overline{\mathbb{R}}:=\widehat{(\mathbb{R},+,d)}$ the Bohr
compactification of $\mathbb{R}$, where $d$ stands for the discrete topology.
This is a compact abelian group, which together with the mapping

$\beta_{\overline{\mathbb{R}}}\colon\mathbb{R}\hookrightarrow\overline{\mathbb{R}},\,\,x\mapsto\left[t\mapsto
e^{-ixt}\right]$

forms a $\omega$-Dirichlet group for every frequency $\omega$.

But for concrete frequencies $\omega$, there often are $\omega$-Dirichlet
groups which in a sense reflect their structure more naturally than the Bohr
compactification.

######  Example 3.4.

Let $\omega=(\log n)$. Then the infinite dimensional torus
$\mathbb{T}^{\infty}$ together with the so-called Kronecker flow

$\beta_{\mathbb{T}^{\infty}}\colon\mathbb{R}\rightarrow\mathbb{T}^{\infty}\,,\,\quad
t\mapsto(\mathfrak{p}_{k}^{-it})\,,$

where $\mathfrak{p}_{k}$ again denotes the $k$th prime, forms a
$\omega$-Dirichlet group. This is basically a consequence of Kronecker’s
approximation theorem (for an alternative ’harmonic analysis’ argument see
[10, Example 3.7]). Note that, identifying

$\mathbb{Z}^{(\mathbb{N})}=\widehat{\mathbb{T}^{\infty}}\,,\quad\alpha\mapsto[z\mapsto
z^{\alpha}]$

($\mathbb{Z}^{(\mathbb{N})}$ all finite sequences of integers), for every
$z\in\mathbb{T}^{\infty}$ one has

(22) $h_{\log n}(z)=h_{\sum\alpha_{j}\log\mathfrak{p}_{j}}(z)=z^{\alpha}\,,$

where $n=\mathfrak{p}^{\alpha}$ with $\alpha\in\mathbb{Z}^{(\mathbb{N})}$.

As announced, we now reformulate the Banach space
$\mathcal{H}_{\infty}^{J}(\omega)$ of all $\omega$-Dirichlet polynomials
supported on the finite index set $J\subset\mathbb{N}$ as a Hardy space of
functions on a $\omega$-Dirichlet group. Given a finite set
$J\subset\mathbb{N}$ and a $\omega$-Dirichlet group $(G,\beta)$, we write

$H_{\infty}^{\omega,J}(G)$

for the subspace of all trigonometric polynomials $f=\sum_{n\in
J}\hat{f}(h_{\omega_{n}})h_{\omega_{n}}$ in $L_{\infty}(G)$ (with respect to
the Haar measure $\mathrm{m}$ on $G$).

The following equivalent description of $\mathcal{H}_{\infty}^{J}(\omega)$ is
now obvious.

######  Proposition 3.5.

Let $J\subset\mathbb{N}$ be a finite subset, and $(G,\beta)$ a
$\omega$-Dirichlet group. Then the Bohr map

$\mathcal{B}:\mathcal{H}_{\infty}^{J}(\omega)\rightarrow
H_{\infty}^{\omega,J}(G),\quad\,\sum_{n\in
J}a_{n}e^{-\omega_{n}s}\mapsto\sum_{n\in J}a_{n}h_{\omega_{n}}$

defines an isometric linear bijection which preserves Bohr and Fourier
coefficients.

###### Proof.

The collections $(e^{-i\omega_{n}\bullet})_{n\in J}$ and
$(h_{\omega_{n}})_{n\in J}$ are both linearly independent. Hence it suffices
to check that for every collection $(a_{n})_{n\in J}$ of complex scalars, we
have

$\sup_{t\in\mathbb{R}}\Big{|}\sum_{n\in
J}a_{n}e^{-i\omega_{n}t}\Big{|}=\sup_{g\in G}\Big{|}\sum_{n\in
J}a_{n}h_{\omega_{n}}(g)\Big{|}\,.$

But this is clear by the fact that
$e^{-i\omega_{n}\bullet}=h_{\omega_{n}}\circ\beta$ for each $n\in J$, and
$\beta$ moreover has dense range. ∎

Finally, we are in the position to prove Theorem 3.1.

###### Proof of Theorem 3.1.

Choose some $\omega$-Dirichlet group $(G,\beta)$ (this is possible by Example
3.3). Then by Proposition 3.5 and Theorem 2.1 one has

$\boldsymbol{\lambda}\big{(}\mathcal{H}_{\infty}^{J}(\omega)\big{)}=\boldsymbol{\lambda}\big{(}H_{\infty}^{\omega,J}(G)\big{)}=\int_{G}\Big{|}\sum_{n\in
J}h_{\omega_{n}}\Big{|}\,d\mathrm{m}\,.$

Applying now Proposition 3.2, we get

$\int_{G}\Big{|}\sum_{n\in
J}h_{\omega_{n}}\Big{|}\,d\mathrm{m}=\lim_{T\to\infty}\frac{1}{2T}\int_{-T}^{T}\Big{|}\sum_{n\in
J}e^{-i\omega_{n}t}\Big{|}\,dt\,,$

so this completes the proof. ∎

We add a ’group version’ of Theorem 3.1, which in view of Proposition 3.2 is
immediate.

######  Theorem 3.6.

Let $J\subset\mathbb{N}$ be a finite subset, $\omega$ a frequency, and
$(G,\beta)$ a $\omega$-Dirichlet group with Haar measure $\mathrm{m}$. Then

$\displaystyle\boldsymbol{\lambda}\big{(}\mathcal{H}_{\infty}^{J}(\omega)\big{)}=\int_{G}\Big{|}\sum_{n\in
J}h_{\omega_{n}}\Big{|}\,d\mathrm{m}\,.$

###  3.2. Projection constants - Dirichlet polynomials

Our aim now is to use the knowledge collected so far, to produce concrete
applications. In fact, we distinguish three cases – each motivated slightly
differently.

####  3.2.1. Case I

We consider Banach spaces $\mathcal{H}_{\infty}^{\leq x}(\omega)$ of Dirichlet
polynomials of length $x$ for the three standard frequencies
$\omega=(n)_{n\in\mathbb{N}_{0}}$, $\omega=(\log p_{n})_{n\in\mathbb{N}}$
(where as above $p_{n}$ is the $n$th prime number), and $\omega=(\log
n)_{n\in\mathbb{N}}$.

Starting with $\omega=(n)_{n\in\mathbb{N}_{0}}$, for $x\in\mathbb{N}$ the
identification

(23) $\mathcal{H}_{\infty}^{\leq
x}(\omega)\,\,=\,\,\text{Trig}_{\\{n\in\mathbb{N}_{0}\colon n\leq
x\\}}(\mathbb{T})\,,\,\,\,\,\,\,\sum_{n=0}^{x}a_{n}e^{-\omega_{n}s}\mapsto\sum_{n=0}^{x}a_{n}z^{n}$

is obviously isometric (doing the variable transformation $z=e^{-s}$), and
hence by Theorem 3.6 we get the integral formula

(24) $\displaystyle\boldsymbol{\lambda}\big{(}\mathcal{H}_{\infty}^{\leq
x}(\omega)\big{)}=\int_{\mathbb{T}}\Big{|}\sum_{n=0}^{x}z^{k}\Big{|}\,dz\,.$

Then a standard calculation leads to the following result, which in view of
(23) is a counterpart of the Lozinski-Kharshiladze theorem from (18).

######  Theorem 3.7.

For the frequency $\omega=(n)_{n\in\mathbb{N}_{0}}$

$\displaystyle\boldsymbol{\lambda}\big{(}\mathcal{H}_{\infty}^{\leq
x}(\omega)\big{)}=\frac{4}{\pi^{2}}\log(x+1)+o(1)\,.$

Next we consider the frequency $\omega=(\log p_{n})$. This is the canonical
example of a $\mathbb{Q}$-linearly independent frequency. It is well-known
(see, e.g., [25, Theorem 9.2]) that, if $\omega_{1},\ldots,\omega_{n}$ are
real numbers, which are linearly independent over $\mathbb{Q}$, and
$\omega_{0}=0$, then for any choice of complex numbers
$a_{0},a_{1},\ldots,a_{x}$ we have

$\sum_{n=0}^{x}|a_{n}|=\sup_{t\in\mathbb{R}}\Big{|}\sum_{n=0}^{x}a_{n}e^{-i\omega_{n}t}\Big{|}\,.$

In other terms, for any frequency $\omega$, which is linearly independent over
$\mathbb{Q}$, and for any finite subset $J\subset\mathbb{N}$ the
identification

$\mathcal{H}_{\infty}^{J}(\omega)\,\,=\,\,\ell_{1}^{|J|}\,,\,\,\quad\sum_{n\in
J}a_{n}e^{-\omega_{n}s}\mapsto(a_{n})_{n\in J}$

is isometric. Then the following integral and limit formula is an immediate
consequences of (4) and (8).

######  Theorem 3.8.

Let $\omega$ be a $\mathbb{Q}$-linearly independent frequency and
$J\subset\mathbb{N}$ a finite subset. Then

$\displaystyle\boldsymbol{\lambda}\big{(}\mathcal{H}_{\infty}^{J}(\omega)\big{)}=\int_{\mathbb{T}^{|J|}}\Big{|}\sum_{n\in
J}z_{k}\Big{|}dz\,.$

Moreover,

$\lim_{x\to\infty}\frac{\boldsymbol{\lambda}\big{(}\mathcal{H}_{\infty}^{\leq
x}(\omega)\big{)}}{\sqrt{x}}=\frac{\sqrt{\pi}}{2}\,.$

We may, despite the precise constant, extend the preceding result within the
setting of $\Lambda(2)$-sets explained in Section 2.2.1. Recall from Remark
2.4 that every $B_{2}$-set of characters on a compact abelian group is a
$\Lambda(2)$-set with $\Lambda(2)$-constant $\leq\sqrt{2}$.

Hence, Theorem 3.6, Corollary 2.3, and Remark 2.4 imply the following
extension of Theorem 3.8.

######  Theorem 3.9.

Let $\omega$ be a frequency and $(G,\beta)$ a $\omega$-Dirichlet group with
the property that all characters $h_{\omega_{n}}\in\widehat{G}$ form a
$\Lambda(2)$-set with constant $C_{2}(\omega)$. Then

$\frac{1}{C_{2}(\omega)}\sqrt{x}\,\,\leq\,\,\boldsymbol{\lambda}\big{(}\mathcal{H}_{\infty}^{\leq
x}(\omega)\big{)}\,\,\leq\,\,\sqrt{x}\,.$

In particular, if $\omega$ satisfies the $B_{2}$-condition, then

$\frac{1}{\sqrt{2}}\sqrt{x}\,\,\leq\,\,\boldsymbol{\lambda}\big{(}\mathcal{H}_{\infty}^{\leq
x}(\omega)\big{)}\,\,\leq\,\,\sqrt{x}\,.$

Finally, we handle the ordinary frequency $\omega=(\log n)$ \- getting one of
our main applications for free. Indeed, combining Harper’s deep result from
(1) with Theorem 3.1, we obtain the precise asymptotic order of the projection
constant of the Banach space $\mathcal{H}_{\infty}^{\leq x}$.

######  Theorem 3.10.

$\displaystyle\boldsymbol{\lambda}\big{(}\mathcal{H}_{\infty}^{\leq
x}\big{(}(\log n)\big{)}\big{)}=O\left(\frac{\sqrt{x}}{(\log\log
x)^{\frac{1}{4}}}\right)$

It seems interesting to rephrase this result again in terms of trigonometric
polynomials. We write

$\Delta(x)=\big{\\{}\alpha\in\mathbb{N}_{0}^{\pi(x)}\colon
1\leq\mathfrak{p}^{\alpha}\leq x\big{\\}}\,,$

where $x\geq 1$ and $\pi(x)$ as usual counts the primes $\mathfrak{p}\leq x$.
Then by Proposition 3.5 we have that

(25) $\mathcal{H}_{\infty}^{\leq x}\big{(}(\log
n)\big{)}=\text{Trig}_{\Delta(x)}\big{(}\mathbb{T}^{\pi(x)}\big{)}\,,$

where the identification is given by the Bohr transform – being isometric and
coefficient preserving. Hence the following two results are immediate
consequences of Theorem 3.10 (first statement) and Theorem 2.1 (second
statement).

######  Corollary 3.11.

$\boldsymbol{\lambda}\big{(}\text{Trig}_{\Delta(x)}\big{(}\mathbb{T}^{\pi(x)}\big{)}\big{)}=O\left(\frac{\sqrt{x}}{(\log\log
x)^{\frac{1}{4}}}\right)\,.$

Moreover,

$\displaystyle\boldsymbol{\lambda}\big{(}\text{Trig}_{\Delta(x)}\big{(}\mathbb{T}^{\pi(x)}\big{)}\big{)}\,=\,\big{\|}\mathbf{P}_{\Delta(x)}:C(\mathbb{T}^{\pi(x)})\rightarrow\text{Trig}_{\Delta(x)}\big{(}\mathbb{T}^{\pi(x)}\big{)}\big{\|}\,=\,\int_{\mathbb{T}^{\pi(x)}}\big{|}\sum_{\alpha\in\Delta(x)}z^{\alpha}\big{|}dz\,,$

where $\mathbf{P}_{\Delta(x)}$ stands for the restriction of the orthogonal
projection on $L_{2}(\mathbb{T}^{\pi(x)})$ onto
$\text{Trig}_{\Delta(x)}\big{(}\mathbb{T}^{\pi(x)}\big{)}$ .

####  3.2.2. Case II

For the ordinary frequency $\omega=(\log n)$ we look at Banach spaces
$\mathcal{H}_{\infty}^{J}(\omega)$ of Dirichlet polynomials generated by index
sets $J$ of natural numbers $k$, which have an a priori specified complexity
of their prime number decompositions. More precisely, the first two results
consider index sets $J$, where each element $k=\mathfrak{p}^{\alpha}\in J$
incorporates not more than $n$ different prime divisors with
$\max\alpha_{j}\leq m$, or similarly, $n$ different prime divisors with
$|\alpha|=\sum\alpha_{j}\leq m$. Recall from (10) the definition of the index
sets $\Lambda_{1}(\leq m,n)$ and $\Lambda_{\infty}(\leq m,n)$.

######  Theorem 3.12.

Let $\omega=(\log n)$. Then

$\boldsymbol{\lambda}\big{(}\mathcal{H}_{\infty}^{N_{\infty}(\leq
m,n)}(\omega)\big{)}\sim(1+\log m)^{n}\,,$

and for each $n$

$\displaystyle\lim_{m\to\infty}\frac{\boldsymbol{\lambda}\big{(}\mathcal{H}_{\infty}^{N_{\infty}(\leq
m,n)}(\omega)\big{)}}{\log^{n}m}=\Big{(}\frac{4}{\pi^{2}}\Big{)}^{n}\,,$

where

$\displaystyle N_{\infty}(\leq
m,n)=\big{\\{}k\in\mathbb{N}\colon\text{$k=\mathfrak{p}^{\alpha}$, \,\,
$\alpha\in\Lambda_{\infty}(\leq m,n)$ }\big{\\}}\,.$

###### Proof.

It follows from Proposition 3.5 and Example 3.4 together with (22) that the
Bohr map

$\mathcal{B}\colon\mathcal{H}_{\infty}^{N_{\infty}(\leq
m,n)}(\omega)\to\text{Trig}_{\Lambda_{\infty}(\leq
m,n)}\big{(}\mathbb{T}^{n}\big{)}\,,\,\,\,\,\,\,\sum_{n\in
J}a_{n}e^{-\omega_{n}s}\mapsto\sum_{\alpha\in\Lambda_{\infty}(\leq
m,n)}a_{\mathfrak{p}^{\alpha}}z^{\alpha}$

defines an isometric linear bijection which preserves Bohr and Fourier
coefficients. Hence the first asymptotic is a consequence of Corollary 2.9
(iv), and the formula for the limit of Corollary 2.10. ∎

Replacing the use of Corollary 2.10 by Corollary 2.5 and (17), similar
arguments lead to the following counterpart of the preceding theorem.

######  Theorem 3.13.

Let $\omega=(\log n)$. Then for each $m,n\in\mathbb{N}$ one has

$\frac{1}{\sqrt{2^{m}}}\left(\sum_{k=0}^{m}\binom{n+k+1}{k}\right)^{\frac{1}{2}}\,\,\,\leq\,\,\,\boldsymbol{\lambda}\big{(}\mathcal{H}^{N_{1}(\leq
m,n)}_{\infty}(\omega)\big{)}\leq\left(\sum_{k=0}^{m}\binom{n+k+1}{k}\right)^{\frac{1}{2}}\,,$

where

$\displaystyle N_{1}(\leq
m,n)=\big{\\{}k\in\mathbb{N}\colon\text{$k=\mathfrak{p}^{\alpha}$, where
$\alpha\in\Lambda_{1}(\leq m,n)$ }\big{\\}}\,.$

We finish this part with a result, which in view of Weissler’s inequality from
(16) is a formal extension of the last theorem. To do so, define for the
finite index set $J\subset\mathbb{N}$, the numbers

$\pi(J)=\max_{n\in J}\,\pi(n)$ and $\Omega(J)=\max_{n\in J}\,\Omega(n)$ ,

and

$\Delta(J)=\big{\\{}\alpha\in\mathbb{N}_{0}^{\pi(J)}\colon n\in
J\,,\,\,\mathfrak{p}^{\alpha}=n\big{\\}}\,.$

Here as usual $\pi(n)$ counts all primes $\leq n$ and $\Omega(n)$ is the
number of prime divisors of $n$ counted according to their multiplicities.

######  Theorem 3.14.

Let $\omega=(\log n)$ and $J\subset\mathbb{N}$ be a finite subset. Then

$\displaystyle\boldsymbol{\lambda}\big{(}\mathcal{H}_{\infty}^{J}(\omega)\big{)}=\int_{\mathbb{T}^{\pi(J)}}\Big{|}\sum_{\alpha\in\Delta(J)}z^{\alpha}\Big{|}\,dz\,.$

Moreover,

$\frac{1}{\sqrt{2^{\Omega(J)}}}\sqrt{|J|}\,\,\,\leq\,\,\,\boldsymbol{\lambda}\big{(}\mathcal{H}_{\infty}^{J}(\omega)\big{)}\leq\sqrt{|J|}.$

###### Proof.

As above (Proposition 3.5), we see that the Bohr map

(26)
$\mathcal{B}\colon\mathcal{H}_{\infty}^{J}(\omega)\to\text{Trig}_{\Delta(J)}\big{(}\mathbb{T}^{\pi(J)}\big{)}\,,\,\,\,\,\,\,\sum_{n\in
J}a_{n}e^{-\omega_{n}s}\mapsto\sum_{\alpha\in\Delta(J)}a_{\mathfrak{p}^{\alpha}}z^{\alpha}$

is isometric and coefficient preserving. Again, the conclusion follows from
Corollary 2.5. Indeed, the order $|\alpha|$ of each multi index
$\alpha\in\Delta(J)$ is at most $\Omega(J)$, and the cardinalities of $J$ and
$\Delta(J)$ are the same. ∎

####  3.2.3. Case III

Finally, we produce an example, where we mix aspects from the preceding two
cases. The following result considers ordinary Dirichlet polynomials of length
$x$ which are supported on all $1\leq n=\mathfrak{p}^{\alpha}\leq x$ having
precisely $m$ prime devisors (all counted according to their multiplicities).

######  Theorem 3.15.

Let $\omega=(\log n)$. Then for each $m\in\mathbb{N}$ one has

$\boldsymbol{\lambda}\big{(}\mathcal{H}_{\infty}^{N_{1}(m,x)}(\omega)\big{)}\,\,\,\sim_{C(m)}\,\,\,\sqrt{\frac{x}{\log
x}}\,\,\,\sqrt{\frac{(\log\log x)^{m-1}}{(m-1)!}}\,,$

where the constants depend on $m$ but not on $x$, and

$N_{1}(m,x)=\big{\\{}1\leq n\leq x\colon
n=\mathfrak{p}^{\alpha}\,\,\,\,\,\text{with}\,\,\,\,\,|\alpha|=m\big{\\}}\,.$

Moreover, the constant $C(m)$ is independent of $m$, whenever
$m\leq\frac{\log\log x}{2e}$.

Let us here first look at the special case $m=1$. Note first that for any $m$
by Proposition 3.5,

$\mathcal{H}_{\infty}^{N_{1}(m,x)}(\omega)=\text{Trig}_{\Delta(N_{1}(m,x))}\big{(}\mathbb{T}^{\pi(N_{1}(m,x))}\big{)}\,,$

where the identification, given by the Bohr map, is isometric and coefficient
preserving. Then for $m=1$ we immediately deduce from Theorem 3.8 and the
prime number theorem that

$\displaystyle\lim_{x\to\infty}\frac{\boldsymbol{\lambda}\big{(}\mathcal{H}_{\infty}^{N_{1}(1,x)}\big{)}}{\sqrt{\frac{x}{\log
x}}}=\lim_{x\to\infty}\frac{\boldsymbol{\lambda}\big{(}\mathcal{H}_{\infty}^{N_{1}(1,x)}\big{)}}{\sqrt{\pi{(x)}}}=\frac{\sqrt{\pi}}{2}\,.$

So the preceding theorem extends this asymptotic estimate from $m=1$ to
arbitrary $m>1$ (neglecting constants).

###### Proof of Theorem 3.15.

Since $\pi(N(m,x))\leq\pi(x)$ and $\Omega(N_{1}(m,x)))=m$, by Theorem 3.14

$\frac{1}{\sqrt{2^{m}}}\sqrt{|N_{1}(m,x)|}\,\,\,\leq\,\,\,\boldsymbol{\lambda}\big{(}\mathcal{H}_{\infty}^{\leq
x,m}\big{)}\leq\sqrt{|N_{1}(m,x)|}\,.$

Then the first claim follows from a well-known result of Landau (see, e.g.,
[45, p. 200]) showing that

$|N_{1}(m,x)|\,\,\,\sim_{C(m)}\,\,\,\frac{x}{\log x}\frac{(\log\log
x)^{m-1}}{(m-1)!}\quad\,\text{as \, $x\to\infty$}\,.$

For the proof of the second claim we use again that by Theorem 3.14

$\boldsymbol{\lambda}\big{(}\mathcal{H}_{\infty}^{N_{1}(m,x)}\big{)}=\int_{\mathbb{T}^{\infty}}\Big{|}\sum_{\alpha\in\Delta(N_{1}(m,x))}z^{\alpha}\Big{|}\,dz\,.$

But in [7, Equation (13), p.107] it is shown that there is a universal
constant $C\geq 1$ such that for any $x,m$ with $m\leq\frac{\log\log x}{2e}$

$\int_{\mathbb{T}^{\infty}}\Big{|}\sum_{\alpha\in\Delta(N_{1}(m,x))}z^{\alpha}\Big{|}\,dz\,\,\,\sim_{C}\,\,\,\Bigg{(}\int_{\mathbb{T}^{\infty}}\Big{|}\sum_{\alpha\in\Delta(N_{1}(m,x))}z^{\alpha}\Big{|}^{2}dz\Bigg{)}^{\frac{1}{2}}\,=\,\sqrt{|N_{1}(m,x)|}\,.$

On the other hand, applying the Sathe-Selberg formula (see, e.g., [12]), we
conclude that for every $\delta>0$ there is $C(\delta)>0$ such that for
$m\leq(2-\delta)\log\log x$

$|N_{1}(m,x)|\,\,\,\sim_{C(\delta)}\,\,\,\frac{x}{\log x}\frac{(\log\log
x)^{m-1}}{(m-1)!}\quad\,\text{as \, $x\to\infty$}\,.$

This completes the argument. ∎

##  4\. Comparing Sidon constants

Recall that, given a topological group $G$, the Sidon constant
$\boldsymbol{\text{\bf Sid}}(\Gamma)$ of a finite set $\Gamma$ of characters
in the dual group $\widehat{G}$ is given by the best constant constant $c\geq
0$ such that for every trigonometric polynomial
$f=\sum_{\gamma\in\Gamma}c_{\gamma}\gamma$ on $G$ one has

(27) $\sum_{\gamma\in\Gamma}|c_{\gamma}|\leq c\|f\|_{\infty}\,.$

It is easily proved that this constant in fact equals the unconditional basis
constant formed by all $\gamma\in\Gamma$ in the Banach space
$\text{Trig}_{\Gamma}(G)$.

Recall that the unconditional basis constant of a basis $(e_{i})_{i\in I}$ of
a Banach space $X$ is given by the infimum over all $K>0$ such that for any
finitely supported family $(\alpha_{i})_{i\in I}$ of scalars and for any
finitely supported family $(\varepsilon_{i})_{i\in I}$ with
$|\varepsilon_{i}|=1,\,i\in I$ we have

$\Big{\|}\sum_{i\in I}\varepsilon_{i}\alpha_{i}e_{i}\Big{\|}\leq
K\Big{\|}\sum_{i\in I}\alpha_{i}e_{i}\Big{\|}\,;$

moreover, in this case the unconditional basis constant of $(e_{i})_{i\in I}$
is defined to be the infimum over all these constants $K$.

In particular, for any finite index set $\Gamma\subset\mathbb{Z}^{n}$ the
unconditional basis constant of the collection of all monomials
$(z^{\alpha})_{\alpha\in\Gamma}$ in $\text{Trig}_{\Gamma}(\mathbb{T}^{n})$,
here denoted by
$\boldsymbol{\chi_{\operatorname{mon}}}\big{(}\text{Trig}_{\Gamma}(\mathbb{T}^{n})\big{)}$,
equals the Sidon constant $\boldsymbol{\text{\bf Sid}}(\Gamma)$.

Given a finite dimensional Banach space $X$, the unconditional basis constant
with respect to a basis of this space and its projection constant are two
quite different objects (compare for example ${\chi}(\ell_{2}^{n})=1$ with
$\boldsymbol{\lambda}(\ell_{2}^{n})\sim\sqrt{n}$).

But in the case of Banach spaces of Dirichlet polynomials and their associated
spaces of multivariate trigonometric polynomials, it turns out that a better
understanding of one of the two constants often leads to a better
understanding of the other constant.

To illustrate this point of view, we start considering analytic trigonometric
polynomials in one variable. Recall that $\text{Trig}_{\\{k\colon 1\leq k\leq
d\\}}(\mathbb{T})$ stands for all analytic trigonometric polynomials of the
form $P(z)=\sum_{k=1}^{d}a_{k}z^{k},\,z~{}\in~{}\mathbb{T}\,,$ so polynomials
of degree $\leq d$ without a constant term $a_{0}$ (following our notation
from Section 2). Then Rudin [38] and Shapiro [44] (see also [9, Proposition
9.7]) proved that

$\frac{1}{\sqrt{2}}\sqrt{d}\leq\boldsymbol{\chi_{\operatorname{mon}}}\big{(}\text{Trig}_{\\{k\colon
1\leq k\leq d\\}}(\mathbb{T})\big{)}\leq\sqrt{d}\,.$

If we allow constant terms, then the best known estimate is

$\sqrt{d}-O(\log
d)^{\frac{2}{3}+\varepsilon}\leq\boldsymbol{\chi_{\operatorname{mon}}}\big{(}\text{Trig}_{\\{k\colon
0\leq k\leq d\\}}(\mathbb{T})\big{)}\leq\sqrt{d}\,.$

This is a deep fact proved by Bombieri and Bourgain in [6], and it shows that
at least from the technical point of view a seemingly small perturbation of
the index set may change the situation drastically.

Let us compare these results with what we in Corollary 2.9 proved for
projection constants, namely

$\boldsymbol{\lambda}\big{(}\text{Trig}_{\\{k\colon 1\leq k\leq
d\\}}(\mathbb{T})\big{)}\sim\boldsymbol{\lambda}\big{(}\text{Trig}_{\\{k\colon
0\leq k\leq d\\}}(\mathbb{T})\big{)}\sim 1+\log d\,.$

This illustrates that unconditional basis constants (so Sidon constants) and
projection constants of spaces of trigonometric polynomials in one variable
behave quite differently.

Of course, the situation doesn’t improve if we admit more variables -
nevertheless we may recognize some in a sense systematic patterns. Indeed, it
was proved in [8] (a result elaborated in [9, Theorem 9.10]) that

$\boldsymbol{\chi_{\operatorname{mon}}}\left(\text{Trig}_{\leq
m}(\mathbb{T}^{n})\right)\sim_{C^{m}}\sqrt{\binom{n+m}{m}}.$

On the other hand by Theorem 3.13 we know that

$\boldsymbol{\lambda}\left(\text{Trig}_{\leq
m}(\mathbb{T}^{n})\right)\sim_{C^{m}}\sqrt{\binom{n+m+1}{m}}\,,$

so in particular we have that

(28) $\boldsymbol{\chi_{\operatorname{mon}}}\left(\text{Trig}_{\leq
m}(\mathbb{T}^{n})\right)\sim_{C^{m}}\boldsymbol{\lambda}\left(\text{Trig}_{\leq
m-1}(\mathbb{T}^{n})\right)\sim_{C^{m}}\left(\frac{n+m}{m}\right)^{\frac{m-1}{2}}\,.$

Coming back to spaces of ordinary(!) Dirichlet polynomials, observe first that
given a finite subset $J$ of $\mathbb{N}$, it is obvious that the ’monomials’
$n^{-s},\,n\in J$ form a basis of the Banach space
$\mathcal{H}_{\infty}^{J}\big{(}(\log)\big{)}$, in the following abbreviated
by $\mathcal{H}_{\infty}^{J}$, and so we denote by

$\boldsymbol{\chi_{\operatorname{mon}}}\big{(}\mathcal{H}_{\infty}^{J}\big{)}$

the unconditional basis constant of all $n^{-s},\,n\in J$.

As already used in (26), there is an isometry
$\mathcal{H}_{\infty}^{J}=\text{Trig}_{\Delta(J)}(\mathbb{T}^{\pi(J)})\,,$
which preserves coefficients, and hence by (28) we conclude that

$\boldsymbol{\chi_{\operatorname{mon}}}\big{(}\mathcal{H}_{\infty}^{N_{1}(m,n)}\big{)}\sim_{C^{m}}\boldsymbol{\lambda}\big{(}\mathcal{H}_{\infty}^{N_{1}(m-1,n)}\big{)}\sim_{C^{m}}\sqrt{\binom{n+m}{m}}\,;$

here we again use the notation fixed in Theorem 3.13.

What about the unconditional basis constants of $\mathcal{H}_{\infty}^{\leq
x}$, the Banach space of all Dirichlet polynomials
$D(s)=\sum_{n=1}^{x}a_{n}n^{-s}$ of length $x$, in comparison with their
projection constants?

Again we use the notation from Section 3.2.2. The following asymptotic

$\boldsymbol{\chi_{\operatorname{mon}}}\big{(}\mathcal{H}_{\infty}^{\leq
x}\big{)}=\boldsymbol{\chi_{\operatorname{mon}}}\big{(}\text{Trig}_{\Delta(x)}(\mathbb{T}^{\pi(x)})\big{)}\sim_{C}\frac{\sqrt{x}}{e^{\big{(}\frac{1}{\sqrt{2}}+o(1)\big{)}\sqrt{\log
x\log\log x}}}\,.$

is taken from [8, Theorem 3] (see also [9, Theorem 9.1]), and it is he final
outcome of a long lasting research project started by Queffélec, Konyagin, and
de la Bretèche (see [9, Section 9.3] for more details on its history). In
Theorem 3.10 and Corollary 3.11 we proved its counterpart for projection
constants:

$\boldsymbol{\lambda}\big{(}\mathcal{H}_{\infty}^{\leq
x}\big{)}=\boldsymbol{\lambda}\big{(}\text{Trig}_{\Delta(x)}(\mathbb{T}^{\pi(x)})\big{)}\sim_{C}\frac{\sqrt{x}}{(\log\log
x)^{\frac{1}{4}}}\,.$

In the $m$-homogeneous case analog results are known - by Balasubramanian-
Calado-Queffélec [3] (a result elaborated in [9, Theorem 9.4]) one has (with
the notation from Theorem 3.15)

$\boldsymbol{{\chi_{\operatorname{mon}}}}\big{(}\mathcal{H}_{\infty}^{N_{1}(m,x)}\big{)}=\boldsymbol{\chi_{\operatorname{mon}}}\big{(}\text{Trig}_{\Delta(N_{1}(m,x))}\big{(}\mathbb{T}^{\pi(N_{1}(m,x))}\big{)}\big{)}\sim_{C(m)}\frac{x^{\frac{m-1}{2m}}}{(\log
x)^{\frac{m-1}{2}}}\,,$

whereas the corresponding result for projection constants from Theorem 3.15
reads

$\boldsymbol{\lambda}\big{(}\mathcal{H}_{\infty}^{N_{1}(m,x)}\big{)}=\boldsymbol{\lambda}\big{(}\text{Trig}_{\Delta(N_{1}(m,x))}\big{(}\mathbb{T}^{\pi(N_{1}(m,x))}\big{)}\big{)}\,\,\,\sim_{C(m)}\,\,\,\sqrt{\frac{x}{\log
x}}\sqrt{\frac{(\log\log x)^{m-1}}{(m-1)!}}\,.$

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