A generalization of K\'atai's orthogonality criterion with applications

TL;DR

This paper generalizes Katai's orthogonality criterion to analyze properties of arithmetic sets from multiplicative number theory, leading to new results on uniform distribution and ergodic sequences for a broad class of functions.

Contribution

It introduces a generalized orthogonality criterion applicable to multiplicative functions, enabling new uniform distribution and ergodic results for level sets of these functions.

Findings

Sequences from level sets of multiplicative functions with positive density are uniformly distributed mod 1 for certain smooth functions.

The generalized criterion applies to a wide class of functions including polynomials with irrational coefficients, fractional powers, and logarithmic functions.

New examples of ergodic sequences are obtained, supporting the ergodic theorem along these sequences.

Abstract

We study properties of arithmetic sets coming from multiplicative number theory and obtain applications in the theory of uniform distribution and ergodic theory. Our main theorem is a generalization of K\'atai's orthogonality criterion. Here is a special case of this theorem: Let $a\colon\mathbb{N}\to\mathbb{C}$ be a bounded sequence satisfying $$ \sum_{n\leq x} a(pn)\overline{a(qn)} = {\rm o}(x),~\text{for all distinct primes $p$ and $q$.} $$ Then for any multiplicative function $f$ and any $z\in\mathbb{C}$ the indicator function of the level set $E=\{n\in\mathbb{N}:f(n)=z\}$ satisfies $$ \sum_{n\leq x} \mathbb{1}_E(n)a(n)={\rm o}(x). $$ With the help of this theorem one can show that if $E=\{n_1<n_2<\ldots\}$ is a level set of a multiplicative function having positive upper density, then for a large class of sufficiently smooth functions $h\colon(0,\infty)\to\mathbb{R}$ the sequence $(h(n_j))_{j\in\mathbb{N}}$ is uniformly distributed $\bmod~1$. This class of functions $h(t)$ includes: all polynomials $p(t)=a_kt^k+\ldots+a_1t+a_0$ such that at least one of the coefficients $a_1,a_2,\ldots,a_k$ is irrational, $t^c$ for any $c>0$ with $c\notin \mathbb{N}$, $\log^r(t)$ for any $r>2$, $\log(\Gamma(t))$, $t\log(t)$, and $\frac{t}{\log t}$. The uniform distribution results, in turn, allow us to obtain new examples of ergodic sequences, i.e. sequences along which the ergodic theorem holds.

Equations

$$\sum _{n⩽x}a(pn)\overline{a(qn)}=\mathrm{o}(x),\text{leavevmode}\text{for all distinct primes p and q.}$$

$$\sum _{n⩽x}\text{mathbbm}{1}_E(n)a(n)=\mathrm{o}(x).$$

$$\sum _{n⩽x}f(n)e(\theta n)=\mathrm{o}(x),$$

$$\underset{N\to \infty }{\lim }\frac{1}{N}\sum _{n=1}^{N}f(\{x_n\})={\int }_0^{1}f(x)dx,$$

$$\sum _{n⩽x}a(pn)\overline{a(qn)}=\mathrm{o}(x),\text{leavevmode}\text{for all distinct primes p and q.}$$

$$\sum _{n⩽x}f(n)a(n)=\mathrm{o}(x).$$

$$E(f,z):=\{n\in \mathbb{N}:f(n)=z\}.$$

$$E(f_1,\dots ,f_r,z_1,\dots ,z_r):=\{n\in \mathbb{N}:f_1(n)=z_1,\dots ,\text{leavevmode}{f}_r(n)=z_r\},$$

$$S_{\Omega ,b_1,r_1}$$

$$S_{\omega ,b_2,r_2}$$

$$S_{\omega ,b_1,r_1}\cap S_{\Omega ,b_2,r_2}=\{n\in \mathbb{N}:\omega (n)\equiv r_1\mathrm{mod}{b}_1,\text{leavevmode}\Omega (n)\equiv r_2\mathrm{mod}{b}_2\}$$

$$S_{\Omega ,\alpha ,J}$$

$$S_{\omega ,\alpha ,J}$$

$$S_{\mathbf{\tau },b,r}$$

$$\sum _{n⩽x}a(pn)\overline{a(qn)}=\mathrm{o}(x),\text{leavevmode}\text{for all distinct primes p and q.}$$

$$\sum _{n⩽x}\text{mathbbm}{1}_E(n)a(n)=\mathrm{o}(x).$$

$$\sum _{n⩽x}\text{mathbbm}{1}_E(n)e(\theta n)=\mathrm{o}(x).$$

$$\underset{N\to \infty }{\lim }\frac{1}{N}\sum _{j=1}^N{T}^{n_j}f={\int }_{X}fd\mu ,$$

$$\mathcal{M}:=\Big{\{}f\nobreak\mskip 2.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\mskip 6.0mu plus 1.0mu\mathbb{N}\to\mathbb{C}:\text{$f$ is multiplicative and}\leavevmode\nobreak\ \sup_{n\in\mathbb{N}}|f(n)|\leqslant 1\Big{\}}.$$

$${\mathbf{\kappa }}_{\xi }(n):=e(\xi \omega (n)),{\mathbf{\lambda }}_{\xi }(n):=e(\xi \Omega (n))$$

$${\mathbf{\mu }}_{\xi }(n):=\{\begin{array}{ll}e(\xi \Omega (n)),& \text{if n is squarefree}\\ 0,& \text{otherwise.}\end{array}$$

$$M(f):=\underset{N\to \infty }{\lim }\frac{1}{N}\sum _{n=1}^{N}f(n).$$

$$M(g)=\prod _{p\in \mathbb{P}}\left(1-\frac{1}{p}\right)\left(1+\sum _{m=1}^{\infty }{p}^{-m}g(p^m)\right).$$

$$\Vert f{\Vert }_1:=\underset{N\to \infty }{\mathrm{limsup}}\frac{1}{N}\sum _{n=1}^N|f(n)|.$$

$$f(n):=e(a(n))=e^{2\pi ia(n)}$$

$$\underset{N\to \infty }{\lim }\frac{1}{N}\sum _{n=1}^{N}F(x(n))={\int }_{\mathbb{T}}Fd\nu .$$

$$\sum _{\genfrac{}{}{0ex}{}{p\in \mathbb{P}}{ma(p)\ne 0\mathrm{mod}1}}\frac{1}{p}=\infty ,\forall m\in \mathbb{N}.$$

$$\underset{N\to \infty }{\lim }\frac{1}{N}\sum _{n=1}^{N}F(x_N(n))={\int }_{\mathbb{T}}Fd\nu .$$

$$a_N(n):=a(n)-\alpha (N),1⩽n⩽N,$$

$$\sum _{\genfrac{}{}{0ex}{}{p\in \mathbb{P}}{ma(p)\ne 0\mathrm{mod}1}}\frac{1}{p}=\infty ,\forall m\in \mathbb{N}.$$

Full text

\sum_{n\leqslant x}a(pn)\overline{a(qn)}={\rm o}(x),\leavevmode\nobreak\ \text{for all distinct primes p$and$q.}

Then for any multiplicative function $f$ and any $z\in\mathbb{C}$ the indicator function of the level set $E=\{n\in\mathbb{N}:f(n)=z\}$ satisfies

$\sum_{n\leqslant x}\mathbbm{1}_{E}(n)a(n)={\rm o}(x).$

With the help of this theorem one can show that if $E=\{n_{1}<n_{2}<\ldots\}$ is a level set of a multiplicative function having positive upper density, then for a large class of sufficiently smooth functions $h:(0,\infty)\to\mathbb{R}$ the sequence $(h(n_{j}))_{j\in\mathbb{N}}$ is uniformly distributed $\bmod\leavevmode\nobreak\ 1$. This class of functions $h(t)$ includes: all polynomials $p(t)=a_{k}t^{k}+\ldots+a_{1}t+a_{0}$ such that at least one of the coefficients $a_{1},a_{2},\ldots,a_{k}$ is irrational, $t^{c}$ for any $c>0$ with $c\notin\mathbb{N}$, $\log^{r}(t)$ for any $r>2$, $\log(\Gamma(t))$, $t\log(t)$, and $\frac{t}{\log t}$. The uniform distribution results, in turn, allow us to obtain new examples of ergodic sequences, i.e. sequences along which the ergodic theorem holds.

1. 1 Introduction

2. 2 Preliminaries

3. 2.1 Multiplicative functions

4. 2.2 Additive functions with values in $\mathbb{T}$

5. 2.3 Additive functions with values in $\mathbb{R}$

6. 3 Extending the Kátai orthogonality criterion

7. 3.1 Definition of $\mathcal{E}_{\text{c.pt.}}^{(\infty)}$ and $\mathcal{E}_{\partial}$

8. 3.2 A generalization of A

9. 3.3 Proof of 3.2

10. 3.4 Proof of 3.5

11. 3.5 Proof of 3.8

12. 3.6 Proof of 3.9

13. 4 Applications to the theory of uniform distribution

14. 5 Applications to Ergodic Theory and proofs of C and F

1.   Introduction

An arithmetic function $f\nobreak\mskip 2.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\mskip 6.0mu plus 1.0mu\mathbb{N}=\{1,2,\ldots,\}\to\mathbb{C}$ is called multiplicative if $f(1)=1$ and $f(mn)=f(m)\cdot f(n)$ for all relatively prime $m,n\in\mathbb{N}$ (and is called completely multiplicative if $f(mn)=f(m)\cdot f(n)$ for all $m,n\in\mathbb{N}$). We start the discussion by formulating the following classical result of Daboussi.

Theorem 1.1** (cf. [7, Theorem 1]).**

Let $f\nobreak\mskip 2.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\mskip 6.0mu plus 1.0mu\mathbb{N}\rightarrow\mathbb{C}$ be a multiplicative function with $|f(n)|\leqslant 1$ for all $n\in\mathbb{N}$. Then for all irrational $\theta$,

[TABLE]

where $e(x):=e^{2\pi ix}$ for all $x\in\mathbb{R}$.

A nice (and shorter) proof of 1.1, which also yields more general results (for instance $e(\theta n)$ replaced with $e(\theta n^{2})$), was later discovered by Kátai [16]. The following theorem is the main technical result that Kátai uses to improve Daboussi’s result and, in addition, to derive new results in the theory of equidistribution (in particular, it is proved in [16] that for any additive function111An arithmetic function $a\nobreak\mskip 2.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\mskip 6.0mu plus 1.0mu\mathbb{N}\to\mathbb{R}$ is called additive if $a(nm)=a(n)+a(m)$ for all $m,n$ with $\gcd(n,m)=1$. $a\nobreak\mskip 2.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\mskip 6.0mu plus 1.0mu\mathbb{N}\to\mathbb{R}$ and any polynomial $p(t)=a_{k}t^{k}+\ldots+a_{1}t+a_{0}$ such that at least one of the coefficients $a_{1},a_{2},\ldots,a_{k}$ is irrational the sequence $a(n)+p(n)$ is *uniformly distributed mod $1$*222A real-valued sequence $(x_{n})_{n\in\mathbb{N}}$ is called uniformly distributed mod $1$ if for all continuous functions $f\nobreak\mskip 2.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\mskip 6.0mu plus 1.0mu[0,1)\to\mathbb{C}$ one has

$\lim_{N\rightarrow\infty}\frac{1}{N}\sum_{n=1}^{N}f(\{x_{n}\})=\int_{0}^{1}f(x)\,dx,$

where for $y\in\mathbb{R}$ the expression $\{y\}$ denotes the fractional part of $y$..).

Theorem 1.2** (Kátai’s orthogonality criterion, see [16, 6]).**

Let $a\nobreak\mskip 2.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\mskip 6.0mu plus 1.0mu\mathbb{N}\to\mathbb{C}$ be a bounded sequence satisfying

[TABLE]

Then for every multiplicative function $f\nobreak\mskip 2.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\mskip 6.0mu plus 1.0mu\mathbb{N}\rightarrow\mathbb{C}$ that is bounded in modulus by $1$, one has

[TABLE]

Given a multiplicative function $f\nobreak\mskip 2.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\mskip 6.0mu plus 1.0mu\mathbb{N}\to\mathbb{C}$ and a point $z\in\mathbb{C}$ let $E(f,z)$ denote the set of solutions to the equation $f(n)=z$, i.e.,

[TABLE]

We will refer to $E(f,z)$ as a level set of $f$. While $E(f,z)$ is defined by means of the multiplicative structure of $\mathbb{N}$, it possesses many interesting properties from the viewpoint of additive integer arithmetic.

Our main result is a generalization of Kátai’s orthogonality criterion in which the multiplicative function $f$ is replaced by the indicator function of a level set of $f$. Actually, our result holds for sets that are more general than sets of the form $E(f,z)$.

Definition 1.3**.**

(Definition of $\mathcal{D}^{(r)}$).

For $r\in\mathbb{N}$ let $\mathcal{D}^{(r)}$ denote the collection of all sets of the from

[TABLE]

where $f_{1},\ldots,f_{r}$ are arbitrary multiplicative functions and $z_{1},\ldots,z_{r}$ are arbitrary complex numbers. It is clear that $\mathcal{D}^{(1)}\subset\mathcal{D}^{(2)}\subset\ldots$; we set $\mathcal{D}^{(\infty)}:=\bigcup_{r=1}^{\infty}\mathcal{D}^{(r)}$.

(Definition of $\mathcal{E}_{\text{c.pt.}}$).

A point $z\in\mathbb{C}$ is called a concentration point for $f\nobreak\mskip 2.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\mskip 6.0mu plus 1.0mu\mathbb{N}\to\mathbb{C}$ if $\sum_{p\leavevmode\nobreak\ \text{prime}\atop f(p)=z}\frac{1}{p}=\infty$ (cf. [17, Definition 3.9]). We define $\mathcal{E}_{\text{c.pt.}}$ to be the collection of all sets of the from $E(f,K):=\{n\in\mathbb{N}:f(n)\in K\}$, where $K$ is an arbitrary subset of $\mathbb{C}$ and $f\nobreak\mskip 2.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\mskip 6.0mu plus 1.0mu\mathbb{N}\to\mathbb{C}$ is a multiplicative function possessing at least one concentration point.

(Definition of $\mathcal{E}_{\text{pol}}$).

A set $K\subset\mathbb{C}$ is an elementary set in polar coordinates if it can be expressed as a finite union of sets of the form $\{re^{2\pi i\varphi}:r\in I_{1},\varphi\in I_{2}\}$, where $I_{1}$ and $I_{2}$ are (open, closed or half-open) intervals in $\mathbb{R}$. Let $\mathcal{E}_{\text{pol}}$ denote the collection of all sets of the form $E(f,K):=\{n\in\mathbb{N}:f(n)\in K\}$, where $K$ is an elementary set in polar coordinates and $f$ is a multiplicative function bounded in modulus by $1$ and satisfying $\lim_{N\to\infty}\frac{1}{N}\sum_{n=1}^{N}|f(n)|\neq 0$ (note that this limit always exists by Wirsing’s mean value theorem, see 2.2 below).

The classes $\mathcal{D}^{(\infty)}$, $\mathcal{E}_{\text{c.pt.}}$ and $\mathcal{E}_{\text{pol}}$ contain numerous classical sets originating in multiplicative number theory. The following (admittedly long) list is comprised of representative examples of sets from these classes which will frequently appear in the next sections of the paper. A more detailed explanation why the sets in Ex.1.4.1 - Ex.1.4.7 below are indeed elements of $\mathcal{D}^{(\infty)}$, $\mathcal{E}_{\text{c.pt.}}$ or $\mathcal{E}_{\text{pol}}$ is provided at the end of Subsection 3.1 (see 3.6).

Example 1.4**.**

1. Ex.1.4.1:

The set $Q$ of squarefree numbers belongs to $\mathcal{D}^{(1)}$. 2. Ex.1.4.2:

Let $\Omega(n)$ denote the number of prime factors of $n$ (counted with multiplicities) and $\omega(n)$ denote the number of distinct prime divisors of $n$ (without multiplicities). For any $b_{1},b_{2},r_{1},r_{2}\in\mathbb{N}$, the sets

[TABLE]

belong to $\mathcal{D}^{(1)}$ and the sets

[TABLE]

belong to $\mathcal{D}^{(2)}$. 3. Ex.1.4.3:

For any irrational $\alpha>0$ and any set $J\subset[0,1)$, the sets

[TABLE]

belong to $\mathcal{E}_{\text{c.pt.}}$ (cf. [11]). 4. Ex.1.4.4:

For any $x\in(0,1)$, the set $\Phi_{x}:=\{n\in\mathbb{N}:\boldsymbol{\varphi}(n)<xn\}$ belongs to $\mathcal{E}_{\text{pol}}$, where $\boldsymbol{\varphi}(n)$ is Euler’s totient function (cf. [18]). 5. Ex.1.4.5:

The set of abundant numbers $\mathscr{A}:=\{n\in\mathbb{N}:\boldsymbol{\sigma}(n)>2n\}$ and the set of deficient numbers $\mathscr{D}:=\{n\in\mathbb{N}:\boldsymbol{\sigma}(n)<2n\}$ belong to $\mathcal{E}_{\text{pol}}$; here $\boldsymbol{\sigma}(n):=\sum_{d\mid n}d$ denotes the sum of divisors function (cf. [8]). 6. Ex.1.4.6:

Let $\boldsymbol{\tau}(n):=\sum_{d\mid n}1$ be the number of divisors function. For $b,r\in\mathbb{N}$ with $\gcd(r,b)=1$, the set

[TABLE]

belongs to $\mathcal{D}^{(t)}$, where $t$ equals the number of generators of the group $(\mathbb{Z}/b\mathbb{Z})^{*}$. More generally, $\{n\in\mathbb{N}:f(n)\equiv r\bmod b\}\in\mathcal{D}^{(t)}$ for any multiplicative function $f\nobreak\mskip 2.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\mskip 6.0mu plus 1.0mu\mathbb{N}\to\mathbb{N}$ (cf. Ex.3.6.2 in Subsection 3.1). 7. Ex.1.4.7:

If $E$ belongs to either $\mathcal{D}^{(\infty)}$, $\mathcal{E}_{\text{c.pt.}}$ or $\mathcal{E}_{\text{pol}}$, then for any multiplicative set333A set $M\subset\mathbb{N}$ is called multiplicative if $1\in M$ and for all $m,n\in\mathbb{N}$ with $\gcd(m,n)=1$ one has $m\cdot n\in M$ if and only if $m\in M$ and $n\in M$. Equivalently, a set $M$ is multiplicative if and only if its indicator function $\mathbbm{1}_{M}$ is a multiplicative function. $M$ the set $E\cap M$ again belongs to $\mathcal{D}^{(\infty)}$, $\mathcal{E}_{\text{c.pt.}}$ or $\mathcal{E}_{\text{pol}}$ respectively. Clearly, any subsemigroup of $(\mathbb{N},\cdot)$ containing $1$ is a multiplicative set. Other examples include the set of $k$-free numbers.

Theorem A** (A generalization of Kátai’s orthogonality criterion).**

Let $a\nobreak\mskip 2.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\mskip 6.0mu plus 1.0mu\mathbb{N}\to\mathbb{C}$ be a bounded sequence satisfying

[TABLE]

If $E\subset\mathbb{N}$ belongs to one of the classes $\mathcal{D}^{(\infty)}$, $\mathcal{E}_{\text{c.pt.}}$ or $\mathcal{E}_{\text{pol}}$ then

[TABLE]

Note that one can quickly derive 1.2 from A. Indeed, any multiplicative function $f\nobreak\mskip 2.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\mskip 6.0mu plus 1.0mu\mathbb{N}\to\mathbb{C}$ that is bounded in modulus by $1$ can be uniformly approximated by finite linear combinations of functions of the form $\mathbbm{1}_{E(f,K)}$, where $K$ is an elementary set in polar coordinates (and hence $E(f,K)\in\mathcal{E}_{\text{pol}}$).

In Section 3 we also state and prove a generalization of A in which the restrictions on $f$ and $K$ in the definition of $\mathcal{E}_{\text{c.pt.}}$ and $\mathcal{E}_{\text{pol}}$ are slightly relaxed (see 3.7). However, the restrictions on $f$ and $K$ in $\mathcal{E}_{\text{c.pt.}}$ and $\mathcal{E}_{\text{pol}}$ cannot be dropped entirely, as there are multiplicative functions $f$ and sets $K\subset\mathbb{C}$ such that (3) does not hold for $E=E(f,K)$444Indeed, if there are no restrictions on $f$ or $K$ then any set $B\subset\mathbb{N}$ can be written in the from $E(f,K)$. Let $(\xi_{n})_{n\in\mathbb{N}}$ be a rationally independent family of irrational numbers in $[0,1)$, let $(p_{n})_{n\in\mathbb{N}}$ be an enumeration of the prime numbers and define $f(p_{1}^{c_{1}}\cdot\ldots\cdot p_{k}^{c_{k}})=e(c_{1}\xi_{1}+\ldots+c_{k}\xi_{k})$. Clearly, $f(n)\neq f(m)$ for all $n\neq m$ and therefore, if we set $K:=\{f(n):n\in B\}$, we get $E(f,K)=B$..

From A, by setting $a(n)=e(n\theta)$, we immediately obtain the following generalization of 1.1.

Corollary B**.**

Suppose $E\subset\mathbb{N}$ belongs to one of the classes $\mathcal{D}^{(\infty)}$, $\mathcal{E}_{\text{c.pt.}}$ or $\mathcal{E}_{\text{pol}}$. Then for any irrational $\theta$ we have

[TABLE]

From B we obtain an application to ergodic theory. We need first the following definition.

Definition 1.5**.**

A sequence $(n_{j})_{j\in\mathbb{N}}$ in $\mathbb{N}$ is called totally ergodic if for any totally ergodic555A measure preserving system $(X,\mathcal{B},\mu,T)$ is called totally ergodic if for every $m\in\mathbb{N}$ the map $T^{m}\nobreak\mskip 2.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\mskip 6.0mu plus 1.0muX\to X$ is ergodic. measure preserving system $(X,\mathcal{B},\mu,T)$ and any $f\in L^{2}$ we have

[TABLE]

where $Tf(x):=f(Tx)$ and the convergence takes place in $L^{2}(X,\mathcal{B},\mu)$.

Using the spectral theorem, it is straightforward to show that a sequence $(n_{j})_{j\in\mathbb{N}}$ is totally ergodic if and only if $(n_{j}\alpha)$ is uniformly distributed mod $1$ for all irrational $\alpha$. Thus B yields the following result.

Corollary C**.**

Let $E=\{n_{1}<n_{2}<\ldots\}$ be a set that belongs to one of the classes $\mathcal{D}^{(\infty)}$, $\mathcal{E}_{\text{c.pt.}}$ or $\mathcal{E}_{\text{pol}}$ and suppose $d(E)$ exists666For any $E\in\mathcal{D}^{(r)}$ it was shown by Ruzsa that the natural density $d(E):=\lim_{N\to\infty}\frac{|E\cap\{1,\ldots,N\}|}{N}$ exists (cf. [17, Corollary 1.6 and the subsequent remark]). The density of sets $E=E(f,K)$ belonging to $\mathcal{E}_{\text{c.pt.}}$ or $\mathcal{E}_{\text{pol}}$ may not exist, but it exists for a rather wide family of sets $E(f,K)$, where the multiplicative function $f$ and the set $K$ are sufficiently regular. In particular, all sets appearing in 1.4 have positive natural density. and is positive. Then $(n_{j})_{j\in\mathbb{N}}$ is a totally ergodic sequence.

A also leads to new uniform distribution results involving functions from Hardy fields. Let $G$ denote the set of all germs777A germ at $\infty$ is an equivalence class of functions under the equivalence relationship (f\sim g)\Leftrightarrow\big{(}\exists t_{0}>0\leavevmode\nobreak\ \text{such that}\leavevmode\nobreak\ f(t)=g(t)\leavevmode\nobreak\ \text{for all}\leavevmode\nobreak\ t\in(t_{0},\infty)\big{)}. at $\infty$ of real valued functions defined on some half-line $(t_{0},\infty)\subset\mathbb{R}$. Note that $G$ forms a ring under pointwise addition and multiplication, which we denote by $(G,+,\cdot)$. Any subfield of the ring $(G,+,\cdot)$ that is closed under differentiation is called a Hardy field. By abuse of language, we say that a function $h\nobreak\mskip 2.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\mskip 6.0mu plus 1.0mu(0,\infty)\to\mathbb{R}$ belongs to some Hardy field $\mathcal{H}$ (and write $f\in\mathcal{H}$) if its germ at $\infty$ belongs to $\mathcal{H}$. See [3, 4, 5] and some references therein for more information on Hardy fields.

Here are some classical examples of functions from Hardy fields.

• •

the class of logarithmico-exponential functions introduced by Hardy in [14, 15], which consists of all functions that can be obtained from polynomials with real coefficients, $\log(t)$ and $\exp(t)$ using the standard arithmetical operations $+$,$-$,$\cdot$,$/$ and the operation of composition (e.g. $\frac{p(t)}{q(t)}$ for all $p,q\in\mathbb{R}[t]$, $t^{c}$ for all $c\in\mathbb{R}$, $\frac{\log t}{t}$, $t\log t$, etc.).

• •

the Gamma function $\Gamma(t)$, the Riemann zeta function $\zeta(t)$, and the logarithmic integral function $\text{Li}(t)$.

Given two functions $f,g\nobreak\mskip 2.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\mskip 6.0mu plus 1.0mu(0,\infty)\to\mathbb{R}$ we write $f(t)\prec g(t)$ if $\frac{g(t)}{f(t)}\to\infty$ as $t\to\infty$. We will say that a function $f(t)$ has polynomial growth if there exists $k\in\mathbb{N}$ such that $f(t)\prec t^{k}$.

The next theorem, which is proved in Section 4, follows from A using elementary computations and results of Boshernitzan [5].

Theorem D**.**

Let $E=\{n_{1}<n_{2}<\ldots\}$ be a set that belongs to either $\mathcal{D}^{(\infty)}$, $\mathcal{E}_{\text{c.pt.}}$ or $\mathcal{E}_{\text{pol}}$. Suppose $h\nobreak\mskip 2.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\mskip 6.0mu plus 1.0mu(0,\infty)\to\mathbb{R}$ belongs to a Hardy field, has polynomial growth and satisfies $|h(t)-r(t)|\succ\log^{2}(t)$ for all polynomials $r\in\mathbb{Q}[t]$. If $d(E)$ exists and is positive then the sequence \big{(}h(n_{j})\big{)}_{j\in\mathbb{N}} is uniformly distributed mod $1$.

In the following corollary we give a sample of particularly interesting cases to which D applies.

Corollary E**.**

Let $E=\{n_{1}<n_{2}<\ldots\}$ be a set that belongs to one of the classes $\mathcal{D}^{(\infty)}$, $\mathcal{E}_{\text{c.pt.}}$ or $\mathcal{E}_{\text{pol}}$ and suppose $d(E)$ exists and is positive. Then

• •

the sequence \big{(}p(n_{j})\big{)}_{j\in\mathbb{N}} is uniformly distributed mod $1$ for any polynomial $p(t)=a_{k}t^{k}+\ldots+a_{1}t+a_{0}$ such that at least one of the coefficients $a_{1},a_{2},\ldots,a_{k}$ is irrational;

• •

the sequence $(n_{j}^{c})_{j\in\mathbb{N}}$ is uniformly distributed mod $1$ for any positive real number $c$ that is not an integer.

• •

the sequence $(\log^{r}n_{j})_{j\in\mathbb{N}}$ is uniformly distributed mod $1$ for any $r>2$.

D also yields applications to ergodic theory.

Definition 1.6** (cf. 1.5 above).**

A sequence $(n_{j})_{j\in\mathbb{N}}$ of integers is called an ergodic sequence if for any ergodic probability measure preserving system $(X,\mathcal{B},\mu,T)$ and any $f\in L^{2}$ we have

[TABLE]

where convergence takes place in $L^{2}(X,\mathcal{B},\mu)$.

Using the spectral theorem and standard techniques in ergodic theory one can derive from D the following corollary.

Corollary F**.**

Let $E=\{n_{1}<n_{2}<\ldots\}$ be a set that belongs to one of the classes $\mathcal{D}^{(\infty)}$, $\mathcal{E}_{\text{c.pt.}}$ or $\mathcal{E}_{\text{pol}}$. Suppose $h\nobreak\mskip 2.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\mskip 6.0mu plus 1.0mu(0,\infty)\to\mathbb{R}$ belongs to a Hardy field, has polynomial growth and satisfies either $\log^{2}t\prec h(t)\prec t$ or $t^{k}\prec h(t)\prec t^{k+1}$ for some $k\in\mathbb{N}$. If $d(E)$ exists and is positive then the sequence \big{(}\lfloor h(n_{j})\rfloor\big{)}_{j\in\mathbb{N}} is an ergodic sequence.

Structure of the paper:

In Section 2 we review basic results and facts regarding multiplicative and additive functions, which are needed in the subsequent sections.

In Section 3 we establish some generalizations of the Kátai orthogonality criterion and, in particular, give a proof of A.

Sections 4 and 5 contain numerous applications of our main results to the theory of uniform distribution and to ergodic theory. D is proved in Section 4 and F is proved in Section 5.

2.   Preliminaries

In this section we present a brief overview of classical results and facts from multiplicative number theory that will be used in subsequent sections.

2.1.   Multiplicative functions

Define

[TABLE]

The following sample amply demonstrates the diversity of multiplicative functions belonging to $\mathcal{M}$; these functions will frequently appear in the later sections.

Example 2.1**.**

1. Ex.2.1.1:

The Liouville function $\boldsymbol{\lambda}$ is defined as $\boldsymbol{\lambda}(n):=(-1)^{\Omega(n)}$ and is completely multiplicative (for the definition of $\Omega(n)$ see 1.4). 2. Ex.2.1.2:

The Möbius function $\boldsymbol{\mu}$ is defined as $\boldsymbol{\mu}(n):=\boldsymbol{\lambda}(n)$ if $n$ is squarefree and $\boldsymbol{\mu}(n):=0$ otherwise. Note that $\boldsymbol{\mu}$ is multiplicative but not completely multiplicative. 3. Ex.2.1.3:

Let $\boldsymbol{\varphi}$ denote Euler’s totient function. Clearly, $\frac{\boldsymbol{\varphi}(n)}{n}\in\mathcal{M}$. 4. Ex.2.1.4:

An arithmetic function $\chi$ is called a Dirichlet character if there exists a number $d\in\mathbb{N}$, called a modulus of $\chi$, such that

1. (1) 

$\chi(n+d)=\chi(n)$ for all $n\in\mathbb{N}$; 2. (2) 

$\chi(n)=0$ whenever $\gcd(d,n)>1$, and $\chi(n)$ is a $\boldsymbol{\varphi}(d)$-th root of unity whenever $\gcd(d,n)=1$; 3. (3) 

$\chi(nm)=\chi(n)\chi(m)$ for all $n,m\in\mathbb{N}$.

Any Dirichlet character is periodic and completely multiplicative. Also $\chi\nobreak\mskip 2.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\mskip 6.0mu plus 1.0mu\mathbb{N}\to\mathbb{C}$ is a Dirichlet character of modulus $k$ if and only if there exists a group character $\widetilde{\chi}$ of the multiplicative group $(\mathbb{Z}/k\mathbb{Z})^{*}$ such that $\chi(n)=\widetilde{\chi}(n\leavevmode\nobreak\ {\rm mod}\leavevmode\nobreak\ k)$ for all $n\in\mathbb{N}$. 5. Ex.2.1.5:

An Archimedean character is a function of the form $n\mapsto n^{it}=e^{it\log n}$ with $t\in\mathbb{R}$. Any Archimedean character is completely multiplicative and takes values in the unit circle. 6. Ex.2.1.6:

Throughout this paper we identify the torus $\mathbb{T}:=\mathbb{R}/\mathbb{Z}$ with the unit interval $[0,1)\bmod 1$ or, when convenient, with the unit circle in the complex plane. Given $\xi\in\mathbb{T}$, let us define the multiplicative functions $\boldsymbol{\kappa}_{\xi}$, $\boldsymbol{\lambda}_{\xi}$ and $\boldsymbol{\mu}_{\xi}$ as

[TABLE]

and

[TABLE]

It is clear that $\boldsymbol{\kappa}_{\xi},\boldsymbol{\lambda}_{\xi},\boldsymbol{\mu}_{\xi}\in\mathcal{M}$.

For $f\in\mathcal{M}$ let $M(f)$ denote the mean value of $f$ whenever it exists, i.e.,

[TABLE]

Note that the mean of a multiplicative function does not always exist (take, for example, Archimedean characters, cf. [12, Section 4.3]).

In the 1960s the study of mean values of multiplicative functions was catalyzed by the works of D’elange, Wirsing and Halász [9, 13, 19]. For real-valued functions in $\mathcal{M}$ Wirsing showed that the mean value always exists:

Theorem 2.2** (Wirsing; see [19] and [10, Theorem 6.4]).**

For any real-valued $g\in\mathcal{M}$ the mean value $M(g)$ exists.

The next theorem is due to Halász [13] and provides easy to check (necessary and sufficient) conditions for $M(g)$ to exist. We use $\mathbb{P}$ to denote the set of prime numbers.

Theorem 2.3** (Halász; see [10, Theorem 6.3]).**

Let $g\in\mathcal{M}$. Then the mean value $M(g)$ exists if and only if one of the following mutually exclusive conditions is satisfied:

1. (i)

there is at least one positive integer $k$ so that $g(2^{k})\neq-1$ and, additionally, the series $\sum_{p\in\mathbb{P}}\frac{1}{p}(1-g(p))$ converges; 2. (ii)

there is a real number $t$ such that $\sum_{p\in\mathbb{P}}\frac{1}{p}(1-{\rm Re}(g(p)p^{it}))$ converges and, moreover, for each positive integer $k$ we have $g(2^{k})=-2^{itk}$; 3. (iii)

$\sum_{p\in\mathbb{P}}\frac{1}{p}(1-{\rm Re}(g(p)p^{it}))=\infty$ for each $t\in\mathbb{R}$.

When condition (i) is satisfied then $M(g)$ is non-zero and can be computed explicitly using the formula

[TABLE]

In the case when $g$ satisfies either (ii) or (iii) then the mean value $M(g)$ equals zero.

Throughout the paper, given a bounded arithmetic function $f\nobreak\mskip 2.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\mskip 6.0mu plus 1.0mu\mathbb{N}\to\mathbb{C}$ we use $\|f\|_{1}$ to denote the seminorm

[TABLE]

Corollary 2.4** (see [2, Lemma 2.9]).**

Suppose $f\in\mathcal{M}$. Then $\|f\|_{1}=0$ if and only if \sum_{p\in\mathbb{P}}\frac{1}{p}\big{(}1-|f(p)|\big{)}=\infty.

Example 2.5**.**

Consider the multiplicative function $\tfrac{\boldsymbol{\varphi}(n)}{n}$ of Ex.2.1.3 on page Ex.2.1.3. By 2.2 we have that $M\left(\frac{\boldsymbol{\varphi}(n)}{n}\right)$ exists. 2.4 implies that $M\left(\frac{\boldsymbol{\varphi}(n)}{n}\right)$ is non-zero. Indeed, $\sum_{p\in\mathbb{P}}\frac{1}{p}\left(1-\frac{\boldsymbol{\varphi}(p)}{p}\right)=\sum_{p\in\mathbb{P}}\frac{1}{p^{2}}<\infty$ and therefore, by 2.4, $\left\|\frac{\boldsymbol{\varphi}(n)}{n}\right\|_{1}>0$. Hence the mean value of $\frac{\boldsymbol{\varphi}(n)}{n}$ is positive.

2.2.   Additive functions with values in $\mathbb{T}$

An arithmetic function $a\nobreak\mskip 2.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\mskip 6.0mu plus 1.0mu\mathbb{N}\to\mathbb{T}$ is called additive if $a(n\cdot m)=a(n)+a(m)\bmod 1$ for all $m,n$ with $\gcd(n,m)=1$. Note that for every additive function $a\nobreak\mskip 2.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\mskip 6.0mu plus 1.0mu\mathbb{N}\to\mathbb{T}$ the function $f\nobreak\mskip 2.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\mskip 6.0mu plus 1.0mu\mathbb{N}\to\{z\in\mathbb{C}:|z|=1\}\subset\mathbb{C}$ defined as

[TABLE]

is a multiplicative function.

Definition 2.6**.**

Let $\nu$ be a Borel probability measure on $\mathbb{T}$ and let $x\nobreak\mskip 2.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\mskip 6.0mu plus 1.0mu\mathbb{N}\to\mathbb{T}$. The sequence $x$ has limiting distribution $\nu$ if for all continuous functions $F\in C(\mathbb{T})$,

[TABLE]

If $\nu$ is the Lebesgue measure on $\mathbb{T}$, then $x(n)$ is said to be uniformly distributed in $\mathbb{T}$.

Theorem 2.7** (see [10, Theorem 8.1, Theorem 8.2 and Remark after Theorem 8.2]).**

Let $a\nobreak\mskip 2.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\mskip 6.0mu plus 1.0mu\mathbb{N}\to\mathbb{T}$ be an additive function and $f(n):=e(a(n))$ denote the corresponding multiplicative function.

1. (a)

The additive function $a(n)$ is uniformly distributed in $\mathbb{T}$ if and only if \sum_{p\in\mathbb{P}}\frac{1}{p}\big{(}1-{\rm Re}(f^{k}(p)p^{it})\big{)}=\infty for all $t\in\mathbb{R}$ and all $k\geqslant 1$. 2. (b)

The additive function $a(n)$ has a limiting distribution $\nu$ that is not the Lebesgue measure if and only if there exists $k\in\mathbb{N}$ such that \sum_{p\in\mathbb{P}}\frac{1}{p}\big{(}1-f^{k}(p)\big{)} converges. The limiting distribution is continuous (i.e. the measure $\nu$ is non-atomic) if and only if

[TABLE]

2.7 gives necessary and sufficient conditions for an additive function to have a limiting distribution. In particular, if an additive function $a(n)$ satisfies neither condition (a) nor condition (b) of 2.7 then $a(n)$ does not possess a limiting distribution. However, even in this case the limiting behavior of $a$ is well understood, as is demonstrated by 2.9 below. In order to formulate 2.9, it will be convenient to introduce first the following variant of 2.6.

Definition 2.8**.**

Let $\nu$ be a Borel probability measure on $\mathbb{T}$ and, for every $N\in\mathbb{N}$, let $x_{N}\nobreak\mskip 2.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\mskip 6.0mu plus 1.0mu\{1,\ldots,N\}\to\mathbb{T}$. Then $(x_{N})_{N\in\mathbb{N}}$ is said to have limiting distribution $\nu$ if for all continuous functions $F\in C(\mathbb{T})$,

[TABLE]

Theorem 2.9** (see [10, Theorem 8.9]).**

Let $a\nobreak\mskip 2.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\mskip 6.0mu plus 1.0mu\mathbb{N}\to\mathbb{T}$ be an additive function. Then there exist $\alpha\nobreak\mskip 2.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\mskip 6.0mu plus 1.0mu\mathbb{N}\to\mathbb{T}$ and a Borel probability measure $\nu$ on $\mathbb{T}$ such that if $a_{N}\nobreak\mskip 2.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\mskip 6.0mu plus 1.0mu\{1,\ldots,N\}\to\mathbb{T}$ denotes the sequence

[TABLE]

then $(a_{N})_{N\in\mathbb{N}}$ has a limiting distribution $\nu$. Moreover, the measure $\nu$ is continuous (i.e. non-atomic) if and only if

[TABLE]

2.3.   Additive functions with values in $\mathbb{R}$

In this subsection we summarize some known results regarding the distribution of real-valued additive functions.

Recall from Footnote 1 that an arithmetic function $a\nobreak\mskip 2.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\mskip 6.0mu plus 1.0mu\mathbb{N}\to\mathbb{R}$ is called additive if $a(n\cdot m)=a(n)+a(m)$ for all $m,n$ with $\gcd(n,m)=1$. For every additive function $a\nobreak\mskip 2.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\mskip 6.0mu plus 1.0mu\mathbb{N}\to\mathbb{R}$, the function

[TABLE]

is a real-valued multiplicative function.

Definition 2.10**.**

Let $\nu$ be a Borel probability measure on $\mathbb{R}$. A sequence $x\nobreak\mskip 2.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\mskip 6.0mu plus 1.0mu\mathbb{N}\to\mathbb{R}$ has limiting distribution $\nu$ if for all bounded continuous functions $F\in C_{b}(\mathbb{R})$,

[TABLE]

Theorem 2.11** (Erdős-Wintner, see [10, Theorem 5.1]).**

An additive function $a\nobreak\mskip 2.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\mskip 6.0mu plus 1.0mu\mathbb{N}\to\mathbb{R}$ possess a limiting distribution if and only if the three series

[TABLE]

converge. In this case the corresponding measure is continuous (i.e. non-atmonic) if and only if

[TABLE]

Corollary 2.12**.**

Let $f\in\mathcal{M}$ be a multiplicative function taking values in $(0,1]$ and assume $\|f\|_{1}\neq 0$. Then $f(n)$ possesses a limiting distribution. This limiting distribution is continuous (i.e. the corresponding measure $\nu$ is non-atomic) if and only if $\sum_{p\in\mathbb{P}\atop f(p)\neq 1}\tfrac{1}{p}=\infty$.

Proof.

Let $a\nobreak\mskip 2.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\mskip 6.0mu plus 1.0mu\mathbb{N}\to\mathbb{R}$ denote the additive function $a(n):=\log(f(n))$. Note that $f$ has a limiting distribution if and only if $a$ has one.

We have $|a(p)|>1$ if and only if $f(p)\in\left(0,\tfrac{1}{e}\right)$. Since $\|f\|_{1}\neq 0$, it follows from 2.4 that \sum_{p\in\mathbb{P}}\frac{1}{p}\big{(}1-f(p)\big{)}<\infty. Therefore

[TABLE]

Also, using the basic inequality $\tfrac{1}{e}(1-x)\geqslant-\log(x)$ for all $x\in\left[\tfrac{1}{e},1\right]$, we obtain

[TABLE]

Therefore, the three series

[TABLE]

converge and hence $a(n)$ possesses a distribution. Clearly, $f$ possesses a continuous distribution if and only if $a$ does, which is the case (by 2.11) if and only if $\sum_{p\in\mathbb{P}\atop|a(p)|>0}\frac{1}{p}=\sum_{p\in\mathbb{P}\atop f(p)\neq 1}\tfrac{1}{p}=\infty$. ∎

3.   Extending the Kátai orthogonality criterion

In Section 1 we introduced the classes $\mathcal{D}^{(\infty)}$, $\mathcal{E}_{\text{c.pt.}}$ and $\mathcal{E}_{\text{pol}}$; the statement of A holds for any set $E$ belonging to either one of these two classes. In this section we will state and prove a generalization of A where $\mathcal{D}^{(\infty)}$, $\mathcal{E}_{\text{c.pt.}}$ and $\mathcal{E}_{\text{pol}}$ are replaced by the more general classes $\mathcal{E}_{\text{c.pt.}}^{(\infty)}$ and $\mathcal{E}_{\partial}$ defined in the next subsection. This generalization is given by 3.7 formulated in Subsection 3.2.

3.1.   Definition of $\mathcal{E}_{\text{c.pt.}}^{(\infty)}$ and $\mathcal{E}_{\partial}$

Let $r\in\mathbb{N}$. A function $\vec{f}=(f_{1},\ldots,f_{r})\nobreak\mskip 2.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\mskip 6.0mu plus 1.0mu\mathbb{N}\to\mathbb{C}^{r}$ is called multiplicative if each of its coordinate components $f_{i}\nobreak\mskip 2.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\mskip 6.0mu plus 1.0mu\mathbb{N}\to\mathbb{C}$ is a multiplicative function. In accordance with the definition of concentration points for multiplicative functions $f\nobreak\mskip 2.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\mskip 6.0mu plus 1.0mu\mathbb{N}\to\mathbb{C}$ (cf. 1.3), we say that a point $\vec{z}\in\mathbb{C}^{r}$ is a concentration point for a multiplicative function $\vec{f}\nobreak\mskip 2.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\mskip 6.0mu plus 1.0mu\mathbb{N}\to\mathbb{C}^{r}$ if the set $P:=\{p\in\mathbb{P}:\vec{f}(p)=\vec{z}\}$ satisfies $\sum_{p\in P}\frac{1}{p}=\infty$.

Definition 3.1**.**

We denote by $\mathcal{E}_{\text{c.pt.}}^{(r)}$ the collection of all sets $E\subset\mathbb{N}$ of the form

[TABLE]

where $K$ is an arbitrary subset of $\mathbb{C}^{r}$ and $\vec{f}\nobreak\mskip 2.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\mskip 6.0mu plus 1.0mu\mathbb{N}\to\mathbb{C}^{r}$ is a multiplicative function possessing at least one concentration point. Observe that $\mathcal{E}_{\text{c.pt.}}=\mathcal{E}_{\text{c.pt.}}^{(1)}$ and $\mathcal{E}_{\text{c.pt.}}^{(i)}\subset\mathcal{E}_{\text{c.pt.}}^{(j)}$ for $i\leqslant j$. We define $\mathcal{E}_{\text{c.pt.}}^{(\infty)}:=\bigcup_{r=1}^{\infty}\mathcal{E}_{\text{c.pt.}}^{(r)}$.

Proposition 3.2**.**

If $E\in\mathcal{D}^{(r)}$ and $d(E)>0$ then $E\in\mathcal{E}_{\text{c.pt.}}^{(r)}$.

A proof of 3.2 will be given in Subsection 3.3.

In order to introduce the class $\mathcal{E}_{\partial}$ we need the following definition.

Definition 3.3**.**

Let $f\nobreak\mskip 2.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\mskip 6.0mu plus 1.0mu\mathbb{N}\to\mathbb{C}$ be an arithmetic function. We define $\mathcal{N}(f)$ – the class of $f$-null sets – to be the collection of all sets $C\subset\mathbb{C}{\backslash}\{0\}$ such that for all $\varepsilon>0$ there exists a continuous function $F\nobreak\mskip 2.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\mskip 6.0mu plus 1.0mu\mathbb{C}\to[0,1]$ satisfying $F(z)=1$ for all $z\in C$ and

[TABLE]

In many cases multiplicative functions have a limiting distribution corresponding to a Borel probability measure $\nu$ (cf. Subsections 2.2 and 2.3). If this is the case then the class of $f$-null sets coincides with the class of $\nu$-null sets, i.e. all sets $C$ that satisfy $\nu(C)=0$. For instance, if $f=\boldsymbol{\lambda}_{\xi}$ for some irrational $\xi\in\mathbb{T}$, then $(\boldsymbol{\lambda}_{\xi}(n))_{n\in\mathbb{N}}$ is uniformly distributed in the unit circle $\mathbb{S}^{1}:=\{z\in\mathbb{C}:|z|=1\}$ (by 2.7 part (a)). It is then straightforward to verify that a set $C\subset\mathbb{C}$ belongs to $\mathcal{N}(\boldsymbol{\lambda}_{\xi})$ if and only if $C\cap\mathbb{S}^{1}$ has zero measure with respect to the Lebesgue measure on $\mathbb{S}^{1}$.

In the following let $\partial J:=\overline{J}{\backslash}J^{\circ}$ denote the boundary of a set $J\subset\mathbb{C}$.

Definition 3.4**.**

1. (a)

Given a multiplicative function $f$ define $\mathcal{A}^{*}(f):=\{J\subset\mathbb{C}{\backslash}\{0\}:\partial J\in\mathcal{N}(f)\}$ and

[TABLE]

It is straightforward to check that both $\mathcal{A}^{*}(f)$ and $\mathcal{A}(f)$ are algebras, i.e. they are closed under finite unions, finite intersections and taking complements. 2. (b)

We denote by $\mathcal{E}_{\partial}$ the collection of all sets $E\subset\mathbb{N}$ of the form $E(f,K):=\{n\in\mathbb{N}:f(n)\in K\}$, where $f\in\mathcal{M}$ with $\|f\|_{1}\neq 0$, and $K\in\mathcal{A}(f)$.

Proposition 3.5**.**

We have $\mathcal{E}_{\text{pol}}\subset\mathcal{E}_{\partial}\cup\mathcal{E}_{\text{c.pt.}}^{(1)}$.

A proof of 3.5 is given in Subsection 3.4.

We will introduce and discuss now two pertinent families of general examples of sets belonging to $\mathcal{E}_{\text{c.pt.}}^{(\infty)}$ and/or $\mathcal{E}_{\partial}$.

Example 3.6**.**

1. Ex.3.6.1:

Let $\alpha_{1},\ldots,\alpha_{t},\beta_{1},\ldots,\beta_{t}$ be real numbers and let $J_{1},\ldots,J_{t},I_{1},\ldots,I_{t}$ be arbitrary subsets of $[0,1)$. Consider the set

[TABLE]

Then $E$ belongs to the class $\mathcal{E}_{\text{c.pt.}}^{(2t)}$ because it can be written as

[TABLE]

where $\boldsymbol{\lambda}_{\xi}$ and $\boldsymbol{\kappa}_{\xi}$ are as defined in Ex.2.1.6 and $J_{i}^{\prime}:=\{e(x):x\in J_{i}\}$ and $I_{i}^{\prime}:=\{e(x):x\in I_{i}\}$. Similarly, one can show that the sets $S_{\Omega,b_{1},r_{1}}$, $S_{\omega,b_{2},r_{2}}$, $S_{\omega,b_{1},r_{1}}\cap S_{\Omega,b_{2},r_{2}}$, $S_{\Omega,\alpha,J}$ and $S_{\omega,\alpha,J}$ from 1.4 belong to $\mathcal{D}^{(1)}$, $\mathcal{D}^{(2)}$ and $\mathcal{E}_{\text{c.pt.}}$ respectively; in particular, they all belong to $\mathcal{E}_{\text{c.pt.}}^{(\infty)}$. 2. Ex.3.6.2:

Let $f\nobreak\mskip 2.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\mskip 6.0mu plus 1.0mu\mathbb{N}\to\mathbb{N}$ be a multiplicative function and let $b,r\in\mathbb{N}$ with $\gcd(b,r)=1$. Let $t$ denote the number of generators of $(\mathbb{Z}/b\mathbb{Z})^{*}$. We claim that the set

[TABLE]

belongs to $\mathcal{D}^{(t)}$. For the proof of this claim, choose $b_{1},b_{2},\ldots,b_{t}\in\mathbb{N}$ with $b=b_{1}\cdot\ldots\cdot b_{t}$ and such that $(\mathbb{Z}/b\mathbb{Z})^{*}$ is isomorphic to $C_{b_{1}}\times\ldots\times C_{b_{t}}$, where $C_{n}$ denotes the finite cyclic group of order $n$. For $i\in\{1,\ldots,t\}$ let $c_{i}$ denote a generator of $C_{b_{i}}$. We can identify $r$ with an element $(c_{1}^{r_{1}},\ldots,c_{t}^{r_{t}})\in C_{b_{1}}\times\ldots\times C_{b_{t}}$, where $r_{i}\in\{0,1,\ldots,b_{i}-1\}$ for all $i\in\{1,\ldots,t\}$. For $i\in\{1,\ldots,t\}$ define $\tilde{\chi}_{i}\nobreak\mskip 2.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\mskip 6.0mu plus 1.0muC_{b_{1}}\times\ldots\times C_{b_{t}}\to\mathbb{C}$ as

[TABLE]

Then $\tilde{\chi}_{i}$ can be identified with a Dirichlet character $\chi_{i}$ of modulus $b$ via the isomorphism $(\mathbb{Z}/b\mathbb{Z})^{*}\cong C_{b_{1}}\times\ldots\times C_{b_{t}}$. It is clear that

[TABLE]

and therefore

[TABLE]

This proves that the set $E$ belongs to $\mathcal{D}^{(t)}$. In particular, by choosing $f=\boldsymbol{\tau}$, we see that the set $S_{\boldsymbol{\tau},b,r}$ from Ex.1.4.6 belongs to $\mathcal{D}^{(t)}$.

3.2.   A generalization of A

In light of Propositions 3.2 and 3.5 it is clear that the following result is a generalization of A.

Theorem 3.7**.**

Let $a\nobreak\mskip 2.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\mskip 6.0mu plus 1.0mu\mathbb{N}\to\mathbb{C}$ be a bounded sequence satisfying

[TABLE]

Then for all sets $E\subset\mathbb{N}$ belonging to either $\mathcal{E}_{\text{c.pt.}}^{(\infty)}$ or $\mathcal{E}_{\partial}$ we have

[TABLE]

For the proof of 3.7 we will need the following proposition.

Proposition 3.8**.**

Let $E\subset\mathbb{N}$ be a set that belongs to either $\mathcal{E}_{\text{c.pt.}}^{(\infty)}$ or $\mathcal{E}_{\partial}$ and suppose $\overline{d}(E)>0$. Then for all $\varepsilon>0$ there exist sets $E_{1}\subset E_{2}\subset\mathbb{C}$ and a subset of prime numbers $P\subset\mathbb{P}$ satisfying:

1. (i)

$\overline{d}(E_{2}{\backslash}E_{1})\leqslant\varepsilon$; 2. (ii)

$\sum_{p\in P}\tfrac{1}{p}=\infty$; 3. (iii)

for all $p\in P$ and $n\in\mathbb{N}$ with $\gcd(n,p)=1$ we have $\mathbbm{1}_{E_{1}}(n)\leqslant\mathbbm{1}_{E}(np)\leqslant\mathbbm{1}_{E_{2}}(n)$.

A proof of 3.8 can be found in Subsection 3.5.

Another key ingredient for proving 3.7 is the following generalization of the Kátai Orthogonality Criterion (1.2), which we believe is of independent interest.

Proposition 3.9**.**

Let $P_{y}$ be a subset of $\mathbb{P}$ with $p\leqslant y$ for all $p\in P_{y}$ and

[TABLE]

If $F$, $G_{1}$, $G_{2}$ and $H$ are bounded real-valued arithmetic functions such that for all $n\in\mathbb{N}$ and $p\in\bigcup_{y}P_{y}$ with $\gcd(n,p)=1$ one has

[TABLE]

and if $(u_{n})$ is a bounded sequence in a Hilbert space $\mathscr{H}$ satisfying

[TABLE]

for all $p,q\in\bigcup_{y}P_{y}$ with $p\neq q$ then

[TABLE]

A proof of 3.9 is given in Subsection 3.6.

At this point we have collected all the tools needed to provide a proof of 3.7.

Proof of 3.7.

Let $a(n)$ be a bounded sequence of complex numbers satisfying (6). Let $E\subset\mathbb{N}$ be a set that belongs to either $\mathcal{E}_{\text{c.pt.}}^{(\infty)}$ or $\mathcal{E}_{\partial}$. If $\overline{d}(E)=0$ then (7) is trivially satisfied. Hence we can assume without loss of generality that $\overline{d}(E)>0$. Let $\varepsilon>0$ be arbitrary. According to 3.8 there exist sets $E_{1}\subset E_{2}\subset\mathbb{C}$ and a set of prime numbers $P\subset\mathbb{P}$ satisfying $\overline{d}(E_{2}{\backslash}E_{1})\leqslant\varepsilon$, $\sum_{p\in P}\tfrac{1}{p}=\infty$, and $\mathbbm{1}_{E_{1}}(n)\leqslant\mathbbm{1}_{E}(np)\leqslant\mathbbm{1}_{E_{2}}(n)$ for all $p\in P$ and $n\in\mathbb{N}$ with $\gcd(n,p)=1$.

Now take $P_{y}:=P\cap[1,y]$, $F:=\mathbbm{1}_{E}$, $G_{1}=\mathbbm{1}_{E_{1}}$, $G_{2}=\mathbbm{1}_{E_{2}}$, $H=1$ and $u_{n}=a(n)$. It follows immediately from $\overline{d}(E_{2}{\backslash}E_{1})\leqslant\varepsilon$ that $\|G_{1}-G_{2}\|_{1}\leqslant\varepsilon$. Also, if $p\in P$ and $\gcd(n,p)=1$, then $G_{1}(n)H(p)\leqslant F(np)\leqslant G_{2}(n)H(p)$. This means we can apply 3.9 to obtain

[TABLE]

Since $\varepsilon>0$ was chosen arbitrarily, this proves the theorem. ∎

We end this subsection with formulating an open question.

Question 3.10**.**

Consider the class $\mathcal{E}_{\text{Jor}}$ of all sets of the form $E(f,K):=\{n\in\mathbb{N}:f(n)\in K\}$, where $f\in\mathcal{M}$ with $\|f\|_{1}>0$ and $K$ is a Jordan measurable subset of $\mathbb{C}$. Observe that $\mathcal{E}_{\text{pol}}\subset\mathcal{E}_{\text{Jor}}$. Can A be extended to the class $\mathcal{E}_{\text{Jor}}$?

3.3.   Proof of 3.2

Before embarking on the proof of 3.2 we need to define and discuss the notion concentrated multiplicative functions (which was introduced by Rusza in [17]).

Definition 3.11** (cf. [17, Definition 3.8 and 3.9]).**

A multiplicative function $f\nobreak\mskip 2.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\mskip 6.0mu plus 1.0mu\mathbb{N}\to\mathbb{C}{\backslash}\{0\}$ is called concentrated if it satisfies

1. (i)

$f$ has at least one concentration point; 2. (ii)

the subgroup of $(\mathbb{C}{\backslash}\{0\},\cdot)$ generated by all concentration points of $f$, which we denote by $\mathbf{G}$, is finite; and 3. (iii)

$\sum_{\begin{subarray}{c}p\in\mathbb{P},\\ f(p)\notin\mathbf{G}\end{subarray}}\frac{1}{p}<\infty.$

Theorem 3.12** (special case of [17, Theorem 3.10]).**

Let $f:\mathbb{N}\to\mathbb{C}{\backslash}\{0\}$ be a multiplicative function. If $f$ is not concentrated then for all $z\in\mathbb{C}{\backslash}\{0\}$ the level set $E(f,z)$ has zero density.

Corollary 3.13** (see [2, Corollary 2.17]).**

Let $f\nobreak\mskip 2.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\mskip 6.0mu plus 1.0mu\mathbb{N}\to\mathbb{C}$ be a multiplicative function and $z\in\mathbb{C}{\backslash}\{0\}$. If $d(E(f,z))>0$ then there exists a concentrated multiplicative function $g\nobreak\mskip 2.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\mskip 6.0mu plus 1.0mu\mathbb{N}\to\mathbb{C}{\backslash}\{0\}$ such that

[TABLE]

Before giving the proof of 3.2 we need the following elementary lemma.

Lemma 3.14**.**

Let $f_{1},\ldots,f_{r}\nobreak\mskip 2.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\mskip 6.0mu plus 1.0mu\mathbb{N}\to\mathbb{C}$ be multiplicative functions and suppose that for every $i\in\{1,\ldots,r\}$ there exists a set of primes $P_{i}\subset\mathbb{P}$ satisfying the following two properties:

1. (i)

$\sum_{p\in\mathbb{P}{\backslash}P_{i}}\frac{1}{p}<\infty$; 2. (ii)

the set $\{f_{i}(p):p\in P_{i}\}$ is finite.

Then there exist $z_{1},\ldots,z_{r}\in\mathbb{C}$ and a set $P\subset\mathbb{P}$ with $\sum_{p\in P}\frac{1}{p}=\infty$ such that $f_{i}(p)=z_{i}$ for all $p\in P$ and all $1\leqslant i\leqslant r$.

Proof.

Let $P^{\prime}:=\bigcap_{i=1}^{r}P_{i}$. Then clearly $\sum_{p\in P}\frac{1}{p}=\infty$. Moreover, $\{(f_{1}(p),\dots,f_{r}(p)):p\in P^{\prime}\}$ is finite, so we get a finite partition of $P$ given by the possible $r$-tuples $(z_{1},\dots,z_{r})$ in the set $\{(f_{1}(p),\dots,f_{r}(p)):p\in P^{\prime}\}$. By the pigeon hole principle, for at least one choice of $(z_{1},\dots,z_{r})$, the set $P=\{p\in P^{\prime}:f_{i}(p)=z_{i},1\leqslant i\leqslant r\}$ satisfies $\sum_{p\in P}\frac{1}{p}=\infty$. ∎

Proof of 3.2.

Let $E\in\mathcal{D}^{(r)}$ with $d(E)>0$ be given. By 1.3, there exist multiplicative functions $f_{1},\ldots,f_{r}\nobreak\mskip 2.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\mskip 6.0mu plus 1.0mu\mathbb{N}\to\mathbb{C}$ and complex numbers $z_{1},\ldots,z_{r}$ such that $E=E(f_{1},\ldots,f_{r},z_{1},\ldots,z_{r})=\{n\in\mathbb{N}:f_{1}(n)=z_{1},\ldots,\leavevmode\nobreak\ f_{r}(n)=z_{r}\}$. Note that $E\subset E(f_{i},z_{i})=\{n\in\mathbb{N}:f_{i}(n)=z_{i}\}$, which implies that $d(E(f_{i},z_{i}))>0$ for all $i\in\{1,\ldots,r\}$.

We now define new multiplicative functions $g_{1},\ldots,g_{r}\nobreak\mskip 2.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\mskip 6.0mu plus 1.0mu\mathbb{N}\to\mathbb{C}$ in the following way: For $i\in\{1,\ldots,r\}$, if $z_{i}=0$, set

[TABLE]

On the other hand, if $z_{i}\neq 0$, we take $g_{i}$ to be the concentrated multiplicative function guaranteed by 3.13. Define $\vec{g}:=(g_{1},\ldots,g_{r})$, $\vec{z}:=(z_{1},\ldots,z_{r})$ and $K:=\{\vec{z}\}$. Observe that

[TABLE]

It thus suffices to show that $E(\vec{g},K)\in\mathcal{E}_{\text{c.pt.}}^{(r)}$.

Note that for every $i\in\{1,\ldots,r\}$ there exists a set of primes $P_{i}\subset\mathbb{P}$, satisfying $\sum_{p\in\mathbb{P}{\backslash}P_{i}}\frac{1}{p}<\infty$, such that $\{g_{i}(p):p\in P_{i}\}$ is finite. In light of 3.14 we can find $w_{1},\ldots,w_{r}\in\mathbb{C}$ and a set of primes $P\subset\mathbb{P}$ with $\sum_{p\in P}\frac{1}{p}=\infty$ such that $g_{i}(p)=w_{i}$ for all $p\in P$ and all $1\leqslant i\leqslant r$. This proves that $\vec{g}$ has a concentration point and hence $E(\vec{g},K)$ belongs to $\mathcal{E}_{\text{c.pt.}}^{(r)}$. ∎

3.4.   Proof of 3.5

In this subsection we give a proof of 3.5. First, we need the following useful lemma.

Lemma 3.15**.**

Let $P\subset\mathbb{P}$ and assume $\sum_{p\in\mathbb{P}{\backslash}P}\tfrac{1}{p}<\infty$. Let $\mathcal{A}$ be an algebra of subsets of $\mathbb{C}$ and suppose that for all $K\in\mathcal{A}$ and all $u\in\mathbb{C}$ the set $uK$ belongs to $\mathcal{A}$. Then for all $f,g\in\mathcal{M}$ that satisfy $f(p)=g(p)$ for all $p\in P$ we have $\mathcal{A}\subset\mathcal{A}(f)$ if and only if $\mathcal{A}\subset\mathcal{A}(g)$.

Proof.

It follows from the definition of $\mathcal{A}(f)$ that the set $K$ belongs to $\mathcal{A}(f)$ if and only if $K{\backslash}\{0\}$ belongs to $\mathcal{A}(f)$ (we will use this fact implicitly later).

Define the sets

[TABLE]

and

[TABLE]

Note that the sets $S_{P}$ and $T_{P}$ are multiplicative, hence $\mathbbm{1}_{S_{P}}$ and $\mathbbm{1}_{T_{P}}$ are multiplicative functions (cf. Footnote 3). Also, $f\cdot\mathbbm{1}_{S_{P}}=g\cdot\mathbbm{1}_{S_{P}}$.

Since any natural number $n$ can be written uniquely as $st$, where $s\in S_{P}$, $t\in T_{P}$ and $\gcd(s,t)=1$, $\mathbb{N}$ can be partitioned into

[TABLE]

where $S_{P}^{(t)}:=\{s\in S_{P}:\gcd(s,t)=1\}$.

We now claim that for all $f\in\mathcal{M}$, $\mathcal{A}\subset\mathcal{A}(f)$ if and only if $\mathcal{A}\subset\mathcal{A}(f\cdot\mathbbm{1}_{S_{P}})$. Note that once we prove this claim, the proof of this lemma is completed, because $f\cdot\mathbbm{1}_{S_{P}}=g\cdot\mathbbm{1}_{S_{P}}$ and therefore $\mathcal{A}\subset\mathcal{A}(f)$ if and only if $\mathcal{A}\subset\mathcal{A}(g)$.

First, assume $\mathcal{A}\subset\mathcal{A}(f)$. Let $K\in\mathcal{A}$ be arbitrary and let $J:=K{\backslash}\{0\}$. Since $J\in\mathcal{A}(f)$, for all $\varepsilon>0$ there exists a continuous function $F\nobreak\mskip 2.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\mskip 6.0mu plus 1.0mu\mathbb{C}\to[0,1]$ such that $F(z)=1$ for all $z\in\partial J$ and

[TABLE]

This, however, implies

[TABLE]

which shows that $J\in\mathcal{A}(f\cdot\mathbbm{1}_{S_{P}})$ and therefore $K\in\mathcal{A}(f\cdot\mathbbm{1}_{S_{P}})$.

Next, assume $\mathcal{A}\subset\mathcal{A}(f\cdot\mathbbm{1}_{S_{P}})$. Again, let $K\in\mathcal{A}$ be arbitrary. Fix $\varepsilon>0$ and let $J:=K{\backslash}\{0\}$. Note that $d(S_{P})=M(\mathbbm{1}_{S_{P}})$ exists (due to 2.2) and $d(S_{P})>0$ because $\sum_{p\in\mathbb{P}{\backslash}P}\tfrac{1}{p}<\infty$ and therefore \sum_{p\in\mathbb{P}}\frac{1}{p}\big{(}1-\mathbbm{1}_{S_{P}}(p)\big{)}<\infty (cf. 2.4). Likewise, $\mathbbm{1}_{S_{P}^{(t)}}$ is a multiplicative function and hence $d(S_{P}^{(t)})=M(\mathbbm{1}_{S_{P}}^{(t)})$ exists (again due to 2.2) and is positive (also by 2.4). Using (15) and the fact that $d(tS_{P}^{(t)})=t^{-1}d(S_{P}^{(t)})$ we obtain

[TABLE]

For every $t\in T_{P}$ with $f(t)\neq 0$ the set $(f(t))^{-1}J\in\mathcal{A}\subset\mathcal{A}(f\cdot\mathbbm{1}_{S_{P}})$. This means that for every $t\in T_{P}$ there exists a continuous function $F_{t}\nobreak\mskip 2.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\mskip 6.0mu plus 1.0mu\mathbb{C}\to[0,1]$ such that $F_{t}(z)=1$ for all z\in\partial\big{(}(f(t))^{-1}J\big{)} and

[TABLE]

Pick $M\geqslant 1$ sufficiently large such that $\sum_{t\in T_{P}\atop t>M}\tfrac{d(S_{P}^{(t)})}{t}\leqslant\tfrac{\varepsilon}{2}$. Define

[TABLE]

Certainly, $F$ is continuous and $F(z)=1$ for all $z\in\partial J$. Moreover,

[TABLE]

Since $\varepsilon>0$ was arbitrary, we conclude that $J\in\mathcal{A}(f)$ and therefore $K\in\mathcal{A}(f)$. ∎

Let $\|x\|$ denote the distance of a real number $x$ to the closest integer. For every $\delta>0$ and every $y\in\mathbb{T}$ define function $F_{y,\delta}\in C(\mathbb{T})$ as

[TABLE]

Lemma 3.16**.**

Let $\nu$ be a Borel probability measure on $\mathbb{T}$ and let $(\nu_{N})_{N\in\mathbb{N}}$ be a sequence of Borel probability measures on $\mathbb{T}$ that converges to $\nu$ in the weak-*-topology (i.e., for all $F\in C(\mathbb{T})$, $\lim_{N\to\infty}\int_{\mathbb{T}}F\,d\nu_{N}=\int_{\mathbb{T}}F\,d\nu$). If $\nu$ is non-atomic then for every $\varepsilon>0$ there exist $\delta>0$ and $N_{0}\in\mathbb{N}$ such that

[TABLE]

for all $y\in\mathbb{T}$ and for all $N\geqslant N_{0}$.

Proof.

Define $I_{\delta}(y):=\int_{\mathbb{T}}F_{y,\delta}\,d\nu$. It is clear that $I_{\delta}$ is a continuous function on $\mathbb{T}$ for every $\delta\in(0,1)$. Also, the family $(I_{\delta})_{\delta\in(0,1)}$ is monotonically decreasing in the sense that $I_{\delta_{1}}(y)\geqslant I_{\delta_{2}}(y)$ for all $y\in\mathbb{T}$ and all $\delta_{1}\geqslant\delta_{2}\in(0,1)$. Since $\nu$ is non-atomic, the functions $F_{y,\delta}(x)$ converge to [math] for $\nu$-almost every $x$. Therefore, by the monotone convergence theorem, $I_{\delta}(y)$ converges to [math] as $\delta\to 0$ for every $y$.

We invoke now the classical Dini theorem, which states that a monotonically decreasing sequence of continuous real-valued functions that converges pointwise to a continuous function convergences uniformly. Therefore $I_{\delta}$ converges to [math] uniformly as $\delta\to 0$.

Fix now some $\varepsilon>0$. Pick $\delta>0$ such that $\sup_{y\in\mathbb{T}}I_{2\delta}(y)<\tfrac{\varepsilon}{2}$. We claim that there exists $N_{0}$ such that for all $N\geqslant N_{0}$ and all $y\in\mathbb{T}$ we have

[TABLE]

Assume that, contrary to our claim, there exists an increasing sequence of natural numbers $(N_{j})_{j\in\mathbb{N}}$ such that for every $j\in\mathbb{N}$ there exists $y_{j}\in\mathbb{T}$ with

[TABLE]

The sequence $(y_{j})_{j\in\mathbb{N}}$ has a convergent subsequence. Hence, by passing to it if necessary, we can assume without loss of generality that $\lim_{j\to\infty}y_{j}$ exists. Let $y\in\mathbb{T}$ denote this limit. It is straightforward to verify that for sufficiently large $j$ we have

[TABLE]

Therefore,

[TABLE]

This contradicts $\int_{\mathbb{T}}F_{y_{j},\delta}\,d\nu_{N_{j}}\geqslant\varepsilon$ for all $j\in\mathbb{N}$. ∎

Lemma 3.17**.**

Suppose $f\in\mathcal{M}$ satisfies $\|f\|_{1}\neq 0$ and $f(n)\neq 0$ for all $n\in\mathbb{N}$. Then

[TABLE]

The following proof of 3.17 was provided by a user with alias Lucia as an answer to a question posted by the third author at http://mathoverflow.net. We gratefully acknowledge Lucia’s help.

Proof of 3.17 (see http://mathoverflow.net/questions/215170).

By replacing $f$ with $|f|$ if necessary, we can ssume without loss of generality that $f$ takes values in $(0,1]$. For $0<\delta<1$ and $k\geqslant 1$ put

[TABLE]

Since $\|f\|_{1}>0$, it follows from 2.4 that \sum_{p\in\mathbb{P}}\frac{1}{p}\big{(}1-f(p)\big{)}<\infty. This shows that $F_{1}(\delta)<\infty$ for every $0<\delta<1$ and so $F_{k}$ is a well defined function for all $k\geqslant 1$. Moreover, since $\sum_{k\geqslant 2}\sum_{p\in\mathbb{P}}\frac{1}{p^{k}}=\sum_{p\in\mathbb{P}}\frac{1}{p(p-1)}<\infty$, the function $F$ is well defined in $(0,1)$.

We claim that $F(\delta)$ converges to zero as $\delta\to 0$. For $0<\delta<1$, let

[TABLE]

We have $F(\delta)=\sum_{p^{k}\in\mathscr{B}_{\delta}}\frac{1}{p^{k}}<\infty$. In particular, $F(1/2)<\infty$ and there exists a finite set $H\subset\mathscr{B}_{\frac{1}{2}}$ such that $\sum_{p^{k}\in\mathscr{B}_{1/2}{\backslash}H}\frac{1}{p^{k}}\leqslant\varepsilon$. Take $0<\delta<\min_{p^{k}\in H}f(p^{k})$. Then $\mathscr{B}_{\delta}\subset\mathscr{B}_{1/2}{\backslash}H$ and therefore $F(\delta)\leqslant\sum_{p^{k}\in\mathscr{B}_{1/2}{\backslash}H}\frac{1}{p^{k}}\leqslant\varepsilon$.

For $0<\delta<1$, let $\mathcal{F}_{\mathscr{B}_{\delta}}$ denote the set of $\mathscr{B}_{\delta}$-free numbers, that is $\mathcal{F}_{\mathscr{B}_{\delta}}:=\mathbb{N}{\backslash}\left(\bigcup_{p^{k}\in\mathscr{B}_{\delta}}p^{k}\mathbb{N}\right)$. It is straightforward to show that

[TABLE]

So,

[TABLE]

Notice that $x\geqslant\exp\left(2\log(\delta)(1-x)\right)$ for any $x\in(\delta,1]$. Moreover, for $n=p_{1}^{k_{1}}\cdots p_{r}^{k_{r}}\in\mathcal{F}_{\mathscr{B}_{\delta}}$, $p_{i}^{k_{i}}\in\mathcal{F}_{\mathscr{B}_{\delta}}$ for $1\leqslant i\leqslant r$, so, in particular, $p_{i}^{k_{i}}\not\in\mathscr{B}_{\delta}$ whence $f(p_{i}^{k_{i}})>\delta$, $1\leqslant i\leqslant r$. Thus, for each $n=p_{1}^{k_{1}}\cdots p_{r}^{k_{r}}\in\mathcal{F}_{\mathscr{B}_{\delta}}$, we have

[TABLE]

So, if $f(n)<\varepsilon$ and $n\in\mathcal{F}_{\mathscr{B}_{\delta}}$, then (19) implies that

[TABLE]

This shows that

[TABLE]

Finally, if we set $\delta=\exp(-\sqrt{-\log(\varepsilon)})$, which goes to zero as $\varepsilon$ goes to zero, then this shows that for $\varepsilon>0$ sufficiently small

[TABLE]

which completes the proof. ∎

We are now ready to give a proof of 3.5.

Proof of 3.5.

Suppose $E$ belongs to $\mathcal{E}_{\text{pol}}$. This means that $E$ is of the form $E(f,K):=\{n\in\mathbb{N}:f(n)\in K\}$, where $f\in\mathcal{M}$ with $\|f\|_{1}\neq 0$ and $K$ is an elementary set in polar coordinates. If $f$ has a concentration point then $E\in\mathcal{E}_{\text{c.pt.}}^{(1)}$ and we are done. Let us therefore assume that $f$ possesses no concentration points. It remains to show that any elementary set in polar coordinates belongs to $\mathcal{A}(f)$, because this implies that $E\in\mathcal{E}_{\partial}$.

Let $f^{\prime}\in\mathcal{M}$ denote the multiplicative function uniquely determined by

[TABLE]

Let $P$ denote the set of all primes $p$ such that $f(p)=f^{\prime}(p)$. Since $\|f\|_{1}\neq 0$, it follows from 2.4 that $\sum_{p\in\mathbb{P}{\backslash}P}\tfrac{1}{p}<\infty$. Therefore, using 3.15, we deduce that $\mathcal{A}(f)$ contains all elementary sets in polar coordinates if and only if $\mathcal{A}(f^{\prime})$ does. We can therefore assume without loss of generality that $f(n)\neq 0$ for all $n\in\mathbb{N}$.

Recall that $e(x):=e^{2\pi ix}$. Now suppose $K:=\{re(\varphi):\varphi\in I_{1},r\in I_{2}\}$, where $I_{1}$ is a subinterval of $\mathbb{T}$ and $I_{2}$ is a subinterval of $[0,1]$. We assume that both $I_{1}$ and $I_{2}$ are closed intervals and remark that for open and half-open intervals the same argument applies. Choose $a_{1},b_{1}\in\mathbb{T}$ such that $I_{1}=[a_{1},b_{1}]$ and $a_{2},b_{2}\in[0,1]$ such that $I_{2}=[a_{2},b_{2}]$.

Let $h(n):=|f(n)|$, $n\in\mathbb{N}$, and let $g(n):=\frac{f(n)}{|f(n)|}$. Clearly, $f=g\cdot h$. Let $a\nobreak\mskip 2.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\mskip 6.0mu plus 1.0mu\mathbb{N}\to\mathbb{T}$ be the (unique) additive function such that $g(n)=e(a(n))$ for all $n\in\mathbb{N}$.

We now distinguish three cases:

1. (i)

$\sum_{p\in\mathbb{P}\atop h(p)\neq 1}\tfrac{1}{p}<\infty$ and $\sum_{p\in\mathbb{P}\atop ma(p)\neq 0\bmod 1}\tfrac{1}{p}<\infty$ for some $m\in\mathbb{N}$; 2. (ii)

$\sum_{p\in\mathbb{P}\atop h(p)\neq 1}\tfrac{1}{p}=\infty$; 3. (iii)

$\sum_{p\in\mathbb{P}\atop ma(p)\neq 0\bmod 1}\tfrac{1}{p}=\infty$ for all $m\in\mathbb{N}$.

In case (i), one of the $m$-th roots of unity is a concentration point of $f$, which contradicts the assumption that $f$ possesses no concentration points. Therefore we only have to deal with cases (ii) and (iii).

In case (ii), $h(n)$ possesses a continuous limiting distribution given by a Borel probability measure $\nu_{2}$ on $[0,1]$ (cf. 2.12). Let $\varepsilon>0$ be arbitrary. Pick a continuous $F_{2}\nobreak\mskip 2.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\mskip 6.0mu plus 1.0mu\mathbb{R}\to[0,1]$ such that $F_{2}(a_{2})=F_{2}(b_{2})=1$ and $\int_{0}^{1}F_{2}\,d\nu_{2}\leqslant\varepsilon$; such a function is guaranteed to exist because $\nu_{2}$ is non-atomic. Define a new function $F\nobreak\mskip 2.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\mskip 6.0mu plus 1.0mu\mathbb{C}\to[0,1]$ as $F(re(\varphi))=F_{2}(r)$. Notice that $F(z)=1$ for all $z\in\partial K$. Moreover,

[TABLE]

Since $\varepsilon>0$ was chosen arbitrarily, this proves that $K\in\mathcal{A}(f)$.

Next, we deal with case (iii). Using 2.9 we can find $\alpha\nobreak\mskip 2.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\mskip 6.0mu plus 1.0mu\mathbb{N}\to\mathbb{T}$ and a probability measure $\nu$ on $\mathbb{T}$ such that if $a_{N}\nobreak\mskip 2.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\mskip 6.0mu plus 1.0mu\{1,\ldots,N\}\to\mathbb{T}$ denotes the sequence

[TABLE]

then $(a_{N})_{N\in\mathbb{N}}$ has limiting distribution $\nu$. Moreover, this limiting distribution is continuous because $\sum_{p\in\mathbb{P}\atop ma(p)\neq 0\bmod 1}\tfrac{1}{p}=\infty$ for all $m\in\mathbb{N}$. Fix $\varepsilon>0$. For $y\in\mathbb{T}$ let $\delta_{y}$ denote the point-mass at $y$. Define

[TABLE]

By definition, the limit of $(\nu_{N})_{N\in\mathbb{N}}$ in the weak-*-topology equals $\nu$. Let $F_{y,\delta}$ be as defined in (17). Using 3.16 we can find $\delta>0$ and $N_{0}\in\mathbb{N}$ such that

[TABLE]

for all $y\in\mathbb{T}$ and for all $N\geqslant N_{0}$. In view of 3.17 we have

[TABLE]

In particular, there exists $\eta>0$ such that

[TABLE]

Let $\tilde{F}\nobreak\mskip 2.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\mskip 6.0mu plus 1.0mu\{re(\varphi):\varphi\in\mathbb{T},r\in[\eta,1]\}\to[0,1]$ denote the function

[TABLE]

Let $F\nobreak\mskip 2.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\mskip 6.0mu plus 1.0mu\mathbb{C}\to[0,1]$ be an arbitrary continuous continuation of $\tilde{F}$ to all of $\mathbb{C}$ that satisfies $F(z)=1$ for all $z\in\partial K$. Then

[TABLE]

Now observe that

[TABLE]

It follows from (20) that

[TABLE]

An analogous argument shows that

[TABLE]

We conclude that

[TABLE]

To summarize, the function $F\nobreak\mskip 2.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\mskip 6.0mu plus 1.0mu\mathbb{C}\to[0,1]$ is continuous, it satisfies $F(z)=1$ for all $z\in\partial K$ and it also satsifies

[TABLE]

Since $\varepsilon>0$ is arbitrary, this proves that $K\in\mathcal{A}(f)$. ∎

3.5.   Proof of 3.8

The purpose of this subsection is to present a proof of 3.8. The proof of 3.8 for the case $E\in\mathcal{E}_{\text{c.pt.}}^{(\infty)}$ is fairly easy and straightforward; the proof for the case $E\in\mathcal{E}_{\partial}$, however, is more complicated and relies on the following lemma.

Lemma 3.18**.**

Let $f\in\mathcal{M}$ with $\|f\|_{1}\neq 0$ and let $K\in\mathcal{A}(f)$. Then for all $\varepsilon>0$ there exist sets $K_{1}\subset K_{2}\subset\mathbb{C}$ and a set of prime numbers $P\subset\mathbb{P}$ satisfying:

• –

$f(p)\neq 0$ for all $p\in P$;

• –

$\sum_{p\in P}\tfrac{1}{p}=\infty$;

• –

$f(p)K_{1}\subset K\subset f(p)K_{2}$ for all $p\in P$;

• –

$\overline{d}(\{n\in\mathbb{N}:f(n)\in K_{2}{\backslash}K_{1}\})\leqslant\varepsilon$.

The proof of 3.18 hinges on two other lemmas, namely Lemmas 3.19 and 3.20, which we state and prove first.

Lemma 3.19**.**

Let $f\in\mathcal{M}$ with $\|f\|_{1}\neq 0$. Then there exists $u\in\mathbb{C}$ with $|u|=1$ such that for all $\delta>0$ the set $P_{u,\delta}:=\{p\in\mathbb{P}:|f(p)-u|<\delta\}$ satisfies $\sum_{p\in P_{u,\delta}}\tfrac{1}{p}=\infty$.

Proof.

Recall that $\mathbb{S}^{1}=\{z\in\mathbb{C}:|z|=1\}$. Suppose that for every $u\in\mathbb{S}^{1}$ there exists some $\delta_{u}>0$ such that $\sum_{p\in P_{u,\delta_{u}}}\tfrac{1}{p}<\infty$. Since $B(u,\delta_{u}):=\{z\in\mathbb{C}:|u-z|<\delta_{u}\}$, $u\in\mathbb{S}^{1}$, is an open cover of the compact set $\mathbb{S}^{1}$, we can find a finite sub-cover. In other words, there exist $u_{1},\ldots,u_{r}\in\mathbb{C}$, $|u_{i}|=1$ for $i=1,\ldots,r$, such that $\bigcup_{i=1}^{r}B(u_{i},\delta_{u_{i}})\supset\mathbb{S}^{1}$. Since $\bigcup_{i=1}^{r}B(u_{i},\delta_{u_{i}})$ is an open set containing $\mathbb{S}^{1}$, there exists some $\delta>0$ such that the set $\{z\in\mathbb{C}:1-\delta<|z|<1+\delta\}$ is contained in $\bigcup_{i=1}^{r}B(u_{i},\delta_{u_{i}})$. Define $P:=\{p\in\mathbb{P}:|f(p)|>1-\delta\}$. Then we have

[TABLE]

One the other hand, it follows from $\|f\|_{1}\neq 0$ and 2.4 that \sum_{p\in\mathbb{P}}\frac{1}{p}\big{(}1-|f(p)|\big{)}<\infty and therefore

[TABLE]

However, $\sum_{p\in\mathbb{P}{\backslash}P}\frac{1}{p}<\infty$ and $\sum_{p\in P}\frac{1}{p}<\infty$ yield a contradiction. ∎

Lemma 3.20**.**

Let $f\in\mathcal{M}$ with $\|f\|_{1}\neq 0$, let $J\subset\mathbb{C}{\backslash}\{0\}$ and assume that $\partial J\in\mathcal{N}(f)$. Then for all $\varepsilon>0$ there exist sets $J_{1}\subset J_{2}\subset\mathbb{C}{\backslash}\{0\}$ and a set of prime numbers $P\subset\mathbb{P}$ satisfying:

• –

$f(p)\neq 0$ for all $p\in P$;

• –

$\sum_{p\in P}\tfrac{1}{p}=\infty$;

• –

$f(p)J_{1}\subset J\subset f(p)J_{2}$ for all $p\in P$;

• –

$\overline{d}(\{n\in\mathbb{N}:f(n)\in J_{2}{\backslash}J_{1}\})\leqslant\varepsilon$.

Proof.

Let $\varepsilon>0$ be arbitrary and let $u\in\mathbb{C}$ be as guaranteed by 3.19. We can find a continuous function $F\nobreak\mskip 2.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\mskip 6.0mu plus 1.0mu\mathbb{C}\to[0,1]$ satisfying $F(z)=1$ for all $z\in\partial J$ and

[TABLE]

Let $D:=\{z\in\mathbb{C}:|z|\leqslant 1\}$ be the unit disc in $\mathbb{C}$. We define a new function $G\nobreak\mskip 2.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\mskip 6.0mu plus 1.0muD\to[0,1]$ as $G(z)=F(uz)$ for all $z\in D$. Note that $G$ has the property that $G(z)=1$ for all $z\in\partial(\overline{u}J)$. Let $S:=\left\{z\in\mathbb{C}{\backslash}\{0\}:G(z)>\tfrac{1}{2}\right\}$ and define $J_{1}:=(\overline{u}J){\backslash}S$ and $J_{2}:=(\overline{u}J)\cup S$. It remains to show that $J_{1}$ and $J_{2}$ have the desired properties.

Since $G$ is uniformly continuous, there exists some $\delta_{0}>0$ such that for all $z,w\in D$

[TABLE]

Take $\delta:=\min\left\{\tfrac{\delta_{0}}{2},\tfrac{1}{2}\right\}$. We claim that

[TABLE]

where $B(0,\delta):=\{z\in\mathbb{C}:|z|<\delta\}$.

We prove (22) by contradiction. Assume there are $w\in J_{1}$ and $z\notin\overline{u}J$ such that $|w-z|<\delta$. Since $w\in\overline{u}J$ and $z\notin\overline{u}J$, there exists a point $y\in\partial(\overline{u}J)$ with $|w-y|<\delta$. Using (21) and the fact that $G(y)=1$ we deduce that $G(w)>\frac{3}{4}$. In particular, $w\in S$. However, this contradicts the fact that $J_{1}\cap S=\emptyset$. The inclusion in (23) can be proved in a similar way.

Let $P_{u,\delta}$ be as in the statement of 3.19 and define $P:=P_{u,\delta}$. Then $\sum_{p\in P}\tfrac{1}{p}=\infty$. Also, for all $p\in P$ we have $|f(p)-u|<\delta$ and therefore $f(p)J_{1}\subset uJ_{1}+B(0,\delta)$. Using (22), we then obtain that $f(p)J_{1}\subset J$. Analogously, using $|f(p)-u|<\delta$ and (23) we get $J\subset f(p)J_{2}$ for all $p\in P$.

It remains to show that $\overline{d}(\{n\in\mathbb{N}:f(n)\in J_{2}{\backslash}J_{1}\})\leqslant\varepsilon$. Take any $p\in P$ that satisfies $\tfrac{1}{p}<\tfrac{\varepsilon}{4}$. Note that

[TABLE]

Using (21) we get that $|F(uf(n))-F(f(p)f(n))|\leqslant\tfrac{\varepsilon}{4}$. Hence,

[TABLE]

Finally,

[TABLE]

This shows that $\overline{d}(\{n\in\mathbb{N}:f(n)\in J_{2}{\backslash}J_{1}\})\leqslant\varepsilon$. ∎

Proof of 3.18.

Let $K\in\mathcal{A}(f)$ and $\varepsilon>0$ be arbitrary and define $J:=K{\backslash}\{0\}$. Since $K\in\mathcal{A}(f)$, $\partial J$ is an $f$-null set ($f$-null sets were defined in 3.3) and therefore, by 3.20, we can find sets $J_{1}\subset J_{2}\subset\mathbb{C}$ and $P\subset\mathbb{P}$ such that:

• –

$f(p)\neq 0$ for all $p\in P$;

• –

$\sum_{p\in P}\tfrac{1}{p}=\infty$;

• –

$f(p)J_{1}\subset J\subset f(p)J_{2}$ for all $p\in P$;

• –

$\overline{d}(\{n\in\mathbb{N}:f(n)\in J_{2}{\backslash}J_{1}\})\leqslant\varepsilon$.

Define

[TABLE]

It is now straightforward to check that $P$, $K_{1}$ and $K_{2}$ satisfy the conclusion of 3.18. ∎

We are now in position to give a proof of 3.8.

Proof of 3.8.

We start with the case $E\in\mathcal{E}_{\text{c.pt.}}^{(\infty)}$ and $\overline{d}(E)>0$. Hence $E$ is of the form $E(\vec{f},K):=\{n\in\mathbb{N}:\vec{f}(n)\in K\},$ where $K$ are arbitrary subsets of $\mathbb{C}^{r}$ and $\vec{f}:=(f_{1},\ldots,f_{r})$ is multiplicative function with at least one concentration point. Hence there exist $\vec{z}=(z_{1},\ldots,z_{r})\in\mathbb{C}^{r}$ and a set of primes $P\subset\mathbb{P}$ with $\sum_{p\in P}\frac{1}{p}=\infty$ and $f_{i}(p)=z_{i}$ for all $p\in P$ and all $1\leqslant i\leqslant r$. Take

[TABLE]

where $\vec{f}\cdot\vec{z}=(f_{1}(n)z_{1},\ldots,f_{r}(n)z_{r})\in\mathbb{C}^{r}$. Note that $E_{2}{\backslash}E_{1}=\emptyset$ and therefore $\overline{d}(E_{2}{\backslash}E_{1})=0$. Also, for all $p\in P$ and $n\in\mathbb{N}$ with $\gcd(n,p)=1$, we have

[TABLE]

This shows that $\mathbbm{1}_{E_{1}}(n)=\mathbbm{1}_{E}(np)=\mathbbm{1}_{E_{2}}(n)$ for all $p\in P$ and $n\in\mathbb{N}$ with $\gcd(n,p)=1$.

Next, we deal with the case $E\in\mathcal{E}_{\partial}$. By the definition of $\mathcal{E}_{\partial}$ there exist $f\in\mathcal{M}$ with $\|f\|_{1}\neq 0$ and $K\in\mathcal{A}(f)$ such that $E=E(f,K)=\{n\in\mathbb{N}:f(n)\in K\}$. According to 3.18, we can find sets $K_{1},K_{2}\subset\mathbb{C}$ and a set of prime numbers $P\subset\mathbb{P}$ satisfying:

1. (1) 

$f(p)\neq 0$ for all $p\in P$; 2. (2) 

$\sum_{p\in P}\tfrac{1}{p}=\infty$; 3. (3) 

$f(p)K_{1}\subset K\subset f(p)K_{2}$ for all $p\in P$; 4. (4) 

$\overline{d}(\{n\in\mathbb{N}:f(n)\in K_{2}{\backslash}K_{1}\})\leqslant\varepsilon$.

Define $E_{1}:=\{n\in\mathbb{N}:f(n)\in K_{1}\}$ and $E_{2}:=\{n\in\mathbb{N}:f(n)\in K_{2}\}$. It follows from property (4) that $\overline{d}(E_{2}{\backslash}E_{1})\leqslant\varepsilon$. Using properties (1) and (3), we deduce that $K_{1}\subset(f(p))^{-1}K\subset K_{2}$ for all $p\in P$. Also, if $p\in P$ and $\gcd(n,p)=1$, then

[TABLE]

It follows that $\mathbbm{1}_{E_{1}}(n)\leqslant\mathbbm{1}_{E}(np)\leqslant\mathbbm{1}_{E_{2}}(n)$ for all $p\in P$ and $n\in\mathbb{N}$ with $\gcd(n,p)=1$, which completes the proof. ∎

3.6.   Proof of 3.9

Before embarking on the proof 3.9 we formulate and prove the following variant of the classical Turán-Kubilius inequality.

Lemma 3.21**.**

Let $P$ be a finite subset of $\mathbb{P}$ and let $w(n):=\sum_{p\in P}\mathbbm{1}_{p\mid n}$, where $\mathbbm{1}_{p\mid n}=1$ if $p\mid n$ and $\mathbbm{1}_{p\mid n}=0$ otherwise, and $m:=\sum_{p\in P}\frac{1}{p}$. Then,

[TABLE]

Proof.

First we expand the left hand side of (24) and get

[TABLE]

where

[TABLE]

Note that $\sum_{n\leqslant x}\mathbbm{1}_{p\mid n}=\frac{x}{p}+{\rm O}(1)$ and hence

[TABLE]

Since $m|P|\leqslant|P|^{2}$, we get $\Sigma^{\prime\prime}=xm^{2}+{\rm O}(|P|^{2}).$

Next observe that $\mathbbm{1}_{p\mid n}\mathbbm{1}_{q\mid n}=\mathbbm{1}_{pq\mid n}$ unless $p=q$. Therefore

[TABLE]

We can estimate $\sum_{n\leqslant x}\mathbbm{1}_{pq\mid n}=\frac{x}{pq}+{\rm O}(1)$ and

[TABLE]

Hence (25) can be written as

[TABLE]

Putting everything together we conclude that

[TABLE]

∎

Proof of 3.9.

In what follows $y=y(x)$ will be a slowly growing function, the conditions for the rate of growth being clear from the context. Instead of showing norm-convergence in (11) we will show that

[TABLE]

Let $u\in\mathscr{H}$ with $\|u\|\leqslant 1$ be arbitrary. We have

[TABLE]

We have used the Cauchy-Schwarz inequality in the last line.

Applying 3.21, we get

[TABLE]

Let us assume that $y=y(x)$ is growing sufficiently slow so that

[TABLE]

Hence

[TABLE]

Note that the cardinality of the set $\{n\leqslant x/p:\gcd(n,p)\neq 1\}$ does not exceed $x/p^{2}$. Since $F$, $G_{1}$, $G_{2}$, $H$ and $u_{n}$ are bounded, it follows from (9) that

[TABLE]

This implies that

[TABLE]

Next, we set $P_{k,y}=P_{y}\cap\{n\in\mathbb{N}:2^{k}\leqslant n<2^{k+1}\}$. Hence

[TABLE]

Combining all of the above we get

[TABLE]

Let $A_{k,y}$ be defined as

[TABLE]

Fixing $k$ and applying the Cauchy-Schwarz inequality again, we get

[TABLE]

Using the prime number theorem to estimate $|P_{k,y}|^{\frac{1}{2}}$ we deduce that

[TABLE]

Combining this with equation (27) and using $|P_{k,y}|\leqslant 2^{k}$ we get

[TABLE]

Finally, if $y=y(x)$ is growing sufficiently slowly then, from (10), we obtain that

[TABLE]

for every $p,q\in P_{y}$ with $p\neq q$. Note that $|P_{k,y}|\leqslant y$ and hence

[TABLE]

Thus the inequality (28) becomes

[TABLE]

Since all the estimates above do not depend on $u$ but only on $\|u\|$, it follows that

[TABLE]

This completes the proof. ∎

4.   Applications to the theory of uniform distribution

Recall (cf. Footnote 2 and 2.6) that a sequence $(x_{n})_{n\in\mathbb{N}}$ of real numbers is uniformly distributed mod $1$ if

[TABLE]

This section is dedicated to proving the following generalization of D.

Theorem 4.1**.**

Let $E=\{n_{1}<n_{2}<\ldots\}$ be a set that belongs to either $\mathcal{E}_{\text{c.pt.}}^{(\infty)}$ or $\mathcal{E}_{\partial}$. Suppose $h\nobreak\mskip 2.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\mskip 6.0mu plus 1.0mu(0,\infty)\to\mathbb{R}$ belongs to a Hardy field, has polynomial growth and satisfies $|h(t)-r(t)|\succ\log^{2}(t)$ for all polynomials $r\in\mathbb{Q}[t]$. If $d(E)$ exists and is positive then the sequence \big{(}h(n_{j})\big{)}_{j\in\mathbb{N}} is uniformly distributed mod $1$.

It follows immediately from Propositions 3.2 and 3.5 that D is a special case of 4.1.

In the proof of 4.1 we will be using the following result of Boshernitzan.

Theorem 4.2** (see [5, Theorem 1.3]).**

Let $\mathcal{H}$ be a Hardy field and assume $h\in\mathcal{H}$ has polynomial growth (i.e. $|h(t)|\prec t^{n}$ for some $n\in\mathbb{N}$). Then $(h(n))_{n\in\mathbb{N}}$ is uniformly distributed $\bmod\leavevmode\nobreak\ 1$ if and only if for every polynomial $r\in\mathbb{Q}[t]$ one has $|h(t)-r(t)|\succ\log(t)$.

We will also need the following lemma.

Lemma 4.3**.**

Let $\mathcal{H}$ be a Hardy field and assume $g\in\mathcal{H}$ satisfies $|g(t)|\succ\log^{2}(t)$. Then, for all $p,q\in\mathbb{N}$ with $p\neq q$,

[TABLE]

Proof.

It suffices to show that for all $c>1$ one has

[TABLE]

because (29) follows quickly from (30) by change of variables. Suppose there exists a constant $c>1$ such that (30) is not satisfied. Remembering that $g(ct)-g(t)$ belongs to a Hardy field, this means that there exist $t_{0}\in(0,\infty)$ and $M>0$ such that

[TABLE]

Define $a:=|g(t_{0})|$ and $b:=M\log(ct_{0})$. It follows that

[TABLE]

However, $|g(t)|\succ\log^{2}(t)$ and hence $|g(c^{n}t_{0})|\succ\log^{2}(c^{n}t_{0})\geqslant b^{\prime}n^{2}$ for some constant $b^{\prime}$. This is a contradiction. ∎

Proof of 4.1.

Let $E=\{n_{1}<n_{2}<\ldots\}$ be a set that belongs to either $\mathcal{E}_{\text{c.pt.}}^{(\infty)}$ or $\mathcal{E}_{\partial}$ and assume $d(E)$ exists and is positive. Let $\mathcal{H}$ be a Hardy field, let $h\in\mathcal{H}$ and suppose $h$ has polynomial growth and satisfies $|h(t)-r(t)|\succ\log^{2}(t)$ for all polynomials $r\in\mathbb{Q}[t]$. We want to show that the sequence \big{(}h(n_{j})\big{)}_{j\in\mathbb{N}} is uniformly distributed mod $1$.

In light of Weyl’s criterion it suffices to show that for all $k\in\mathbb{Z}{\backslash}\{0\}$ the averages

[TABLE]

converge to [math] as $N\to\infty$. Since $d(E)$ exits and is positive, this is equivalent to

[TABLE]

In view of 3.7, to prove (31) it suffices to show that

[TABLE]

for all primes $p\neq q$.

We claim that the sequence $(h(pn)-h(qn))_{n\in\mathbb{N}}$ is uniformly distributed mod $1$. Once we have verified this claim, (32) follows immediately, because $\int_{0}^{1}e(kx)\,dx=0$.

Note that $h(pt)-h(qt)$ belongs itself to a Hardy field. According to 4.2, $(h(pn)-h(qn))_{n\in\mathbb{N}}$ is uniformly distributed mod $1$ if and only if for all $r\in\mathbb{Q}[t]$,

[TABLE]

Let $r(t)=c_{k}t^{k}+\ldots+c_{1}t+c_{0}\in\mathbb{Q}[t]$ be arbitrary. Note that the value of $c_{0}$ has no influence on (33) and we can assume that $c_{0}=0$. Define a new polynomial $s(t):=b_{k}t^{k}+\ldots+b_{1}t$, where $b_{i}:=\tfrac{c_{i}}{p^{i}-q^{i}}$, $1\leqslant i\leqslant k$. A simple calculation shows that $r(t)=s(pt)-s(qt)$. Define $g(t):=h(t)-s(t)$. Then (33) can be written as

[TABLE]

However, since $s(t)\in Q[t]$, we have that $|g(t)|=|h(t)-s(t)|\succ\log^{2}(t)$ by our assumption. Therefore (34) follows directly 4.3. This completes the proof. ∎

5.   Applications to Ergodic Theory and proofs of C and F

We start by recalling the following well-known characterizations of ergodic and totally ergodic sequences (see Definitions 1.6 and 1.5).

Theorem 5.1**.**

Let $(n_{j})_{j\in\mathbb{N}}$ be a sequence in $\mathbb{N}$.

1. (a)

The sequence $(n_{j})_{j\in\mathbb{N}}$ is ergodic if and only if for all $\alpha\in\mathbb{R}{\backslash}\mathbb{Z}$,

[TABLE] 2. (b)

The sequence $(n_{j})_{j\in\mathbb{N}}$ is totally ergodic if and only if for all $\alpha\in\mathbb{R}{\backslash}\mathbb{Q}$,

[TABLE]

(It is not hard to see that both parts of 5.1 follow immediately from the spectral theorem.)

5.1 allows us to derive the following corollary from 3.7.

Corollary 5.2**.**

Let $E=\{n_{1}<n_{2}<\ldots\}$ be a set that belongs to either $\mathcal{E}_{\text{c.pt.}}^{(\infty)}$ or $\mathcal{E}_{\partial}$ and suppose $d(E)$ exists and is positive. Then $(n_{j})_{j\in\mathbb{N}}$ is a totally ergodic sequence.

Proof.

It follows from part (b) of 5.1 that it suffices to show that

[TABLE]

for all irrational $\alpha$. Since $d(E)$ exists and is positive, equation (35) is equivalent to

[TABLE]

However, (36) follows from 3.7 because for any irrational $\alpha$ the sequence $e(n\alpha)$ satisfies (6). ∎

Note that in view of Propositions 3.2 and 3.5, C follows directly from 5.2. We also have the following generalization of F.

Theorem 5.3**.**

Let $E=\{n_{1}<n_{2}<\ldots\}$ be a set that belongs to either $\mathcal{E}_{\text{c.pt.}}^{(\infty)}$ or $\mathcal{E}_{\partial}$. Suppose $h\nobreak\mskip 2.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\mskip 6.0mu plus 1.0mu(0,\infty)\to\mathbb{R}$ belongs to a Hardy field $\mathcal{H}$, has polynomial growth and satisfies either $\log^{2}t\prec h(t)\prec t$ or $t^{k}\prec h(t)\prec t^{k+1}$ for some $k\in\mathbb{N}$. If $d(E)$ exists and is positive then \big{(}\lfloor h(n_{j})\rfloor\big{)}_{j\in\mathbb{N}} is an ergodic sequence.

Proof of 5.3 (cf. [1, Lemma 5.12]).

In view of 5.1, part (a), it suffices to show that for every $\alpha\in\mathbb{R}{\backslash}\mathbb{Z}$ we have

[TABLE]

We have $\lfloor h(n)\rfloor=h(n)-\{h(n)\}$. Therefore e\big{(}\lfloor h(n)\rfloor\alpha\big{)}=g\big{(}\alpha h(n_{j}),h(n_{j})\big{)}, where $g\nobreak\mskip 2.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\mskip 6.0mu plus 1.0mu\mathbb{R}^{2}\to\mathbb{C}$ is the function g(x,y)=e\big{(}x-\alpha\{y\}\big{)}. Note that $g$ is $1$-periodic and hence can be viewed as a function from $\mathbb{T}^{2}$ to $\mathbb{C}$. It thus suffices to show that

[TABLE]

Let $H:=\overline{\{(\alpha t\bmod 1,t\bmod 1):t\in\mathbb{R}\}}$. Note that $H$ is a closed subgroup of $\mathbb{T}^{2}$ and one has $H=\mathbb{T}^{2}$ if $\alpha$ is irrational and $H\subsetneq\mathbb{T}^{2}$ if $\alpha$ is rational.

Let $\mu_{H}$ denote the (normalized) Haar measure on $H$. We claim that $\int g\,d\mu_{H}=0$. If $H=\mathbb{T}^{2}$ then $\int g\,d\mu_{H}=\int\left(\int g(x,y)\,dx\right)\,dy=\int 0\,dy=0$. If $H\subsetneq\mathbb{T}^{2}$, then $\alpha$ must be rational and hence

[TABLE]

Therefore,

[TABLE]

for all continuous $f\nobreak\mskip 2.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\mskip 6.0mu plus 1.0muH\to\mathbb{C}$. (Indeed, the left hand side of (38) describes an invariant probability measure on $H$ and any invariant probability measure must coincide with $\mu_{H}$, by uniqueness of Haar measures.) Thus, we have

[TABLE]

Since $\int g\,d\mu_{H}=0$ and $g$ is Riemann integrable, to show (37) it suffices to show that the sequence \big{(}\alpha h(n_{j}),h(n_{j})\big{)}_{j\in\mathbb{N}} is uniformly distributed in $H$. Since any group character of $H$ comes from a character on $\mathbb{T}^{2}$ and the non-trivial characters of $H$ are described by $\{(x,y)\mapsto e(\ell x+my):\ell,m\in\mathbb{Z},\leavevmode\nobreak\ \alpha\ell+m\neq 0\}$, it follows from Weyl’s equdistribution criterion that \big{(}\alpha h(n_{j}),h(n_{j})\big{)}_{j\in\mathbb{N}} is uniformly distributed in $H$ if and only if for all $(\ell,m)\in\mathbb{Z}^{2}$ that satisfy $\alpha\ell+m\neq 0$ one has

[TABLE]

Since $h\in\mathcal{H}$ has polynomial growth and satisfies $n^{k-1}\prec h(t)\prec n^{k}$, we conclude that $(\ell\alpha+m)h(n)$ also belongs to $\mathcal{H}$, has polynomial growth and satisfies $|(\ell\alpha+m)h(t)-r(t)|\succ\log^{2}(t)$ for all $r\in\mathbb{Q}[t]$. It follows from 4.1 that the sequence \big{(}(\ell\alpha+m)h(n_{j})\big{)}_{j\in\mathbb{N}} is uniformly distributed mod $1$. This implies that

[TABLE]

and we conclude that (39) holds. ∎