Correlations of multiplicative functions along deterministic and independent sequences

TL;DR

This paper investigates the correlations of multiplicative functions along both deterministic and independent sequences, employing ergodic theory to extend recent results and analyze structural properties of measure-preserving systems.

Contribution

It introduces a novel ergodic theory approach to study multiplicative functions along sequences, extending previous work by Tao, Teräväinen, and the author.

Findings

Extended correlations results for multiplicative functions

Utilized ergodic theory to analyze deterministic sequences

Applied multiple ergodic theorems to independent sequences

Abstract

We study correlations of multiplicative functions taken along deterministic sequences and sequences that satisfy certain linear independence assumptions. The results obtained extend recent results of Tao and Ter\"av\"ainen and results of the author. Our approach is to use tools from ergodic theory in order to effectively exploit feedback from analytic number theory. The results on deterministic sequences crucially use structural properties of measure preserving systems associated with bounded multiplicative functions that were recently obtained by the author and Host. The results on independent sequences depend on multiple ergodic theorems obtained using the theory of characteristic factors and qualitative equidistribution results on nilmanifolds.

Equations

$${\mathbb{E}}_{m\in \mathbb{N}}\lambda (m+n_1)\cdots \lambda (m+n_{\ell })=0.$$

$${\mathbb{E}}_{m\in \mathbb{N}}^{\log }\lambda (a(m+n_1))\cdots \lambda (a(m+n_{\ell }))=0.$$

$$\lim_{N\to\infty}{\mathbb{E}}_{n_{1},\ldots,n_{\ell}\in[N]}\big{|}{\mathbb{E}}_{m\in{\mathbf{M}}}\,\lambda(m)\,\lambda(m+n_{1})\cdots\lambda(m+n_{\ell})\big{|}=0.$$

$$\lim_{|n|\to\infty,n\in S}\big{(}\mathbb{E}^{\log}_{m\in{\mathbf{M}}}\,\lambda(m)\,\lambda(m+n_{1})\cdots\lambda(m+n_{\ell})\big{)}=0$$

$$\lim_{n\to\infty}\big{(}\mathbb{E}^{\log}_{m\in{\mathbf{M}}}\,\lambda(m)\,\lambda(m+a_{1}(n))\cdots\lambda(m+a_{\ell}(n))\big{)}=0.$$

$${\mathbb{E}}_{n\in{\mathbb{N}}}\big{(}\mathbb{E}^{\log}_{m\in{\mathbf{M}}}\,f_{1}(m+[n\alpha_{1}])\cdots f_{\ell}(m+[n\alpha_{\ell}])\big{)}=\mathbb{E}^{\log}_{m\in{\mathbf{M}}}f_{1}(m)\cdots\mathbb{E}^{\log}_{m\in{\mathbf{M}}}f_{\ell}(m).$$

$${\mathbb{E}}_{n\in A}a(n):=\frac{1}{|A|}\sum _{n\in A}a(n),{\mathbb{E}}_{n\in A}^{\log }a(n):=\frac{1}{{\sum }_{n\in A}\frac{1}{n}}\sum _{n\in A}\frac{a(n)}{n}.$$

$${\mathbb{E}}_{n\in A}a(n):=\underset{N\to \infty }{\lim }{\mathbb{E}}_{n\in A\cap [N]}a(n),{\mathbb{E}}_{n\in A}^{\log }a(n):=\underset{N\to \infty }{\lim }{\mathbb{E}}_{n\in A\cap [N]}^{\log }a(n)$$

$${\mathbb{E}}_{n\in {\mathbb{N}}^r}a(n):=\underset{N\to \infty }{\lim }{\mathbb{E}}_{n\in [N{]}^r}a(n)$$

$${\mathbb{E}}_{n\in \mathbf{M}}a(n):=\underset{k\to \infty }{\lim }{\mathbb{E}}_{n\in [M_k]}a(n),{\mathbb{E}}_{n\in \mathbf{M}}^{\log }a(n):=\underset{k\to \infty }{\lim }{\mathbb{E}}_{n\in [M_k]}^{\log }a(n)$$

$$\underset{N\to \infty }{\lim }\frac{|Z\cap [N{]}^r|}{N^r}=0.$$

$$f(mn)=f(m)f(n)\text{ whenever }(m,n)=1.$$

$${\mathbb{E}}_{p\in \mathbb{P}}^{\log }(1-\mathrm{Re}(a(p)\cdot \overline{b(p)}))=0.$$

$${\mathbb{E}}_{m\in \mathbb{N}}^{\log }\prod _{j=1}^{\ell }{f}_j(a(m+n_j))=0.$$

$$\sup_{n\in{\mathbb{N}}}\Big{|}\sum_{k=1}^{n}f(a(k))\Big{|}=+\infty.$$

$$\lim_{|n|\to\infty,n\in S}\big{(}\mathbb{E}^{\log}_{m\in{\mathbf{M}}}\,\prod_{j=0}^{\ell}f_{j}(m+n_{j})\big{)}=0.$$

$$\lim_{|n|\to\infty}\big{(}\mathbb{E}^{\log}_{m\in{\mathbf{M}}}\prod_{j=0}^{\ell}f_{j}(m+a_{j}(n))\big{)}=0.$$

$$\operatorname{D-lim}_{|n|\to\infty}\big{(}\mathbb{E}^{\log}_{m\in{\mathbf{M}}}\prod_{j=0}^{\ell}f_{j}(m+a_{j}(n))\big{)}=0.$$

$${\mathbb{E}}_{n\in {\mathbb{N}}^r}{\mathbb{E}}_{m\in \mathbf{M}}^{\log }\prod _{j=0}^{\ell }{f}_j(m+a_j(n))=\prod _{j=1}^{\ell }{\mathbb{E}}_{m\in \mathbf{M}}^{\log }{f}_j(m).$$

$$\{m\in \mathbb{N}:f(a(m))={ϵ}_0,f(a(m+n_1))={ϵ}_1,f(a(m+n_2))={ϵ}_2,f(a(m+n_3))={ϵ}_3\}$$

$$d_{\mathbf{M}}(\Lambda ):={\mathbb{E}}_{m\in \mathbf{M}}^{\log }{1}_{\Lambda }(m),$$

$${\Lambda }_{n,ϵ}:=\{m\in \mathbb{N}:f_0(m)={ϵ}_0,f_1(m+a_1(n))={ϵ}_1,\dots ,f_{\ell }(m+a_{\ell }(n))={ϵ}_{\ell }\}.$$

$$\underset{|n|\to \infty }{\lim }(d_{\mathbf{M}}({\Lambda }_{n,ϵ}))=2^{-(\ell +1)}.$$

$${D-lim}_{|n|\to \infty }(d_{\mathbf{M}}({\Lambda }_{n,ϵ}))=2^{-(\ell +1)}.$$

$$\lambda (m)={ϵ}_0,\lambda (m+a_1(n))={ϵ}_1,\dots ,\lambda (m+a_{\ell }(n))={ϵ}_{\ell }.$$

$${\mathbb{E}}_{n\in \mathbb{N}}^{\log }\lambda (a(n))w(n)=0$$

$${\mathbb{E}}_{m\in \mathbf{M}}^{\log }\prod _{j=1}^s{g}_j(a(m+n_j))={\mathbb{E}}_{m\in \mathbf{M}}^{\log }\prod _{j=1}^s{g}_j(m+n_j)$$

$${\mathbb{E}}_{n\in \mathbb{N}}^{\log }\prod _{j=1}^{\ell }{f}_j(a_j(n))=\prod _{j=1}^{\ell }{\mathbb{E}}_{n\in \mathbb{N}}^{\log }{f}_j(n).$$

$$\underset{k\to \infty }{\lim }{\mathbb{E}}_{m\in [M_k]}^{\log }\prod _{j=1}^s{\tilde{a}}_j(m+n_j)$$

$${\mathbb{E}}_{m\in \mathbf{M}}^{\log }\prod _{j=1}^{\ell }{a}_j(m+n_j)=\int \prod _{j=1}^{\ell }{T}^{n_j}{F}_{j}d\mu$$

Full text

The author was supported by the Hellenic Foundation for Research and Innovation, Project No: 1684.

1. Introduction and main results

Let $\lambda\colon{\mathbb{N}}\to\{-1,+1\}$ be the Liouville function that is defined to be $1$ on integers with an even number of prime factors, counted with multiplicity, and $-1$ elsewhere. Its values are expected to be randomly distributed and based on this several conjectures have been formulated. One such conjecture, by Chowla [3], asserts that the values of $\lambda$ form a normal sequence of $\pm 1$. Equivalently, if $n_{1},\ldots,n_{\ell}\in{\mathbb{N}}$ are distinct, then (see Section 1.1 for our notation on averages)

[TABLE]

The conjecture is settled for $\ell=1$, this is elementarily equivalent to the prime number theorem, and remains open for $\ell\geq 2$. Recently, a version involving logarithmic averages was established for $\ell=2$ by Tao [23] and for all odd values of $\ell$ by Tao and Teräväinen [25, 26]. Similar results are also known for Cesàro averages on almost all scales [27]. But even in its logarithmic form, the conjecture remains open for all even $\ell\geq 4$.

It is also expected that if $a\colon{\mathbb{N}}\to{\mathbb{N}}$ is low complexity strictly increasing sequence, then the sequence $\lambda\circ a$ inherits the randomness properties of $\lambda$. It is indeed a classical result of Kamae [19] and Weiss [29], proved in the 70’s, that normality of a sequence is preserved by composition with a deterministic sequence. But since normality of the Liouville function is unknown, it is unclear how to extend known results about correlations of $\lambda$ to results about $\lambda\circ a$. Our first goal is to solve this problem for a large class of deterministic sequences. In particular, it is a consequence of Theorem 1.1 below, that if the sequence $a\colon{\mathbb{N}}\to{\mathbb{N}}$ is deterministic and totally ergodic (for example, take $a(n)=[n\alpha+\beta]$ where $\alpha>1$ is irrational and $\beta\in{\mathbb{R}}$), then for $\ell=2$ and for all odd $\ell$ we have for all distinct $n_{1},\ldots,n_{\ell}\in{\mathbb{N}}$ that

[TABLE]

On a slightly different direction, Matomäki, Radziwiłł, and Tao [22] established an averaged version of the Chowla conjecture, implying that if ${\mathbf{M}}:=(M_{k})_{k\in{\mathbb{N}}}$, with $M_{k}\to\infty$, is such that all limits ${\mathbb{E}}_{m\in{{\mathbf{M}}}}$ below exist, then

[TABLE]

In [4] this result was extended to shifts given by arbitrary linearly independent polynomials $p_{1},\ldots,p_{\ell}\colon{\mathbb{N}}^{r}\to{\mathbb{Z}}$ with zero constant terms. Our second goal is to show that if we replace in (1) the average ${\mathbb{E}}_{m\in{\mathbf{M}}}$ with a logarithmic average, then we can establish results for vastly more general classes of shifts and we can also replace the limit in density with a regular limit. For instance, it follows from Theorem 1.3 that if $S$ is a subset of ${\mathbb{N}}^{\ell}$ with independent elements (see definition in Section 1.3), then

[TABLE]

where $n_{1},\ldots,n_{\ell}$ denote the coordinates of $n\in{\mathbb{N}}^{\ell}$. One corollary of this result is Theorem 1.4, which implies that if the sequences $a_{1},\ldots,a_{\ell}\colon{\mathbb{N}}\to{\mathbb{N}}$ have different growth rates and ${\mathbf{M}}:=(M_{k})_{k\in{\mathbb{N}}}$, with $M_{k}\to\infty$, is such that all limits $\mathbb{E}^{\log}_{m\in{{\mathbf{M}}}}$ below exist, then

[TABLE]

Moreover, we show that the previous results apply to arbitrary collections of multiplicative functions with values on the complex unit disc as long as at least one of them satisfies some aperiodicity assumptions.

Our last goal is to establish related results for correlations of arbitrary multiplicative functions $f_{1},\ldots,f_{\ell}\colon{\mathbb{N}}\to[-1,1]$. It follows from Theorem 1.5 below that if $\alpha_{1},\ldots,\alpha_{\ell}\in{\mathbb{R}}$ are rationally independent, and ${\mathbf{M}}$ is as before, then

[TABLE]

Note that this identity is no longer true if we replace the average ${\mathbb{E}}_{n\in{\mathbb{N}}}$ with a limit or a limit in density, or if the multiplicative functions $f_{1},\ldots,f_{\ell}$ take values on the complex unit disc. On the other hand, we show that the shifts can be replaced by arbitrary collections of sequences that satisfy some linear independence and equidistribution assumptions.

1.1. Definitions and notation

In order to facilitate exposition, we introduce some definitions and notation.

1.1.1. Averages

For $N\in{\mathbb{N}}$ we let $[N]:=\{1,\dots,N\}$. Let $a\colon{\mathbb{N}}\to{\mathbb{C}}$ be a sequence. If $A$ is a non-empty finite subset of ${\mathbb{N}}$ we let

[TABLE]

If $A$ is an infinite subset of ${\mathbb{N}}$ we let

[TABLE]

if the limits exist. Also, if $a\colon{\mathbb{N}}^{r}\to{\mathbb{C}}$ is a sequence, we let

[TABLE]

if the limit exists.

Lastly, let ${\mathbf{M}}=([M_{k}])_{k\in{\mathbb{N}}}$ be a sequence of intervals with $M_{k}\to\infty$. We let

[TABLE]

if the limits exist. Henceforth, we implicitly assume that the lengths of all sequences of intervals considered increase to infinity.

1.1.2. Convergence in density

A subset $Z$ of ${\mathbb{N}}^{r}$ has (natural) density [math] if

[TABLE]

If $n=(n_{1},\ldots,n_{r})\in{\mathbb{N}}^{r}$, we let $|n|:=|n_{1}|+\cdots+|n_{r}|$.

Definition**.**

Let $r\in{\mathbb{N}}$. We say that the sequence $a\in\ell^{\infty}({\mathbb{N}}^{r})$ converges in density to [math], and write $\operatorname{D-lim}_{|n|\to\infty}a(n)=0$, if any of the following three equivalent conditions hold:

1. (i)

For every $\varepsilon>0$ the set $\{n\in{\mathbb{N}}^{r}\colon|a(n)|\geq\varepsilon\}$ has natural density [math]; 2. (ii)

$\lim_{N\to\infty}{\mathbb{E}}_{n\in[N]^{r}}|a(n)|=0$; 3. (iii)

$\lim_{|n|\to\infty,n\notin Z\,}a(n)=0$ for some $Z\subset{\mathbb{N}}^{r}$ with natural density [math].

Several other notions used in the next two subsections are properly defined in Section 2.

1.2. Correlations along deterministic sequences

We start by giving some results related to correlations of multiplicative functions composed with a fixed low complexity sequence. The following definition gives precise meaning to the term low complexity and also defines the notion of total ergodicity that is crucial for our purposes (see Section 2 for basic background in ergodic theory and for the definition of F-systems).

Definition**.**

With ${\mathbb{U}}$ we denote the complex unit disc. We say that:

• •

A sequence $a\colon{\mathbb{N}}\to{\mathbb{U}}$ is totally ergodic if all its F-systems are totally ergodic. It is deterministic if all its F-systems have zero entropy.

• •

A sequence $a\colon{\mathbb{N}}\to{\mathbb{N}}$ is totally ergodic (or deterministic) if it is strictly increasing, its range $A:=a({\mathbb{N}})$ is a set of positive density, and the $\{0,1\}$-valued sequence ${\bf 1}_{A}$ is totally ergodic (respectively, deterministic).

Remarks**.**

$\bullet$ In the bibliography (for example in [19, 29, 30]) a deterministic sequence is often referred to as completely deterministic. Note that these definitions use Cesàro averages, but for our purposes in the definition of F-systems we use logarithmic averages.

$\bullet$ See [1, Lemma 4.25] (or [30]) for necessary and sufficient conditions for a sequence $a\colon{\mathbb{N}}\to\mathbb{U}$ to be deterministic that involve the word complexity of the sequence. For finite valued sequences, when deterministic sequences are defined using Cesàro averages, they read as follows: For every $\varepsilon>0$ there exists $N\in{\mathbb{N}}$ such that if we change the values of $a(n)$ on a set of $n\in{\mathbb{N}}$ with natural density at most $\varepsilon$, then we can get a new sequence that has at most $2^{\varepsilon N}$ words of length $N$ on its range.

An example of a sequence $a\colon{\mathbb{N}}\to{\mathbb{N}}$ that is deterministic and totally ergodic is $a(n)=n$ and another one is $a(n)=[n\alpha+\beta]$ where $\alpha$ is an irrational greater than $1$ and $\beta\in{\mathbb{R}}$. Other examples can be given by considering a uniquely and totally ergodic system $(X,\mu,T)$ with zero entropy and constructing $a\colon{\mathbb{N}}\to{\mathbb{N}}$ by taking the elements of the set $S=\{n\in{\mathbb{N}}\colon T^{n}x_{0}\in U\}$ in increasing order, where $x_{0}\in X$ is arbitrary and $U$ is a set with positive measure and with boundary of measure zero. One such example is the set $S:=\{n\in{\mathbb{N}}\colon\{n^{d}\alpha\}\in[b,c)\}$ where $d\in{\mathbb{N}}$, $0\leq b<c<1$, and $\alpha$ is irrational (for $d=1$ these examples include all sequences of the form $[n\alpha+\beta]$, where $\alpha>1$ is irrational).

Definition**.**

A function $f\colon{\mathbb{N}}\to{\mathbb{C}}$ is called multiplicative if

[TABLE]

It is called completely multiplicative if the previous identity holds for every $m,n\in{\mathbb{N}}$. A Dirichlet character is a periodic completely multiplicative function $\chi$ with $\chi(1)=1$. For convenience, we extend all multiplicative functions to ${\mathbb{Z}}$ by letting $f(n)=0$ for $n\leq 0$.

Henceforth, with ${\mathbb{P}}$ we denote the set of prime numbers. For notational convenience we use the following notion of equivalence:

Definition**.**

Let $a,b\colon{\mathbb{P}}\to\mathbb{U}$. We write $a\sim b$ if

[TABLE]

Remarks**.**

$\bullet$ If we restrict to sequences that take values on the unit circle, then $\sim$ is an equivalence relation and $a\sim b$ is equivalent to $\mathbb{E}^{\log}_{p\in{\mathbb{P}}}|a(p)-b(p)|^{2}=0$.

$\bullet$ Using terminology from [26] we have that two multiplicative functions $f,g\colon{\mathbb{N}}\to\mathbb{U}$ satisfy $f\sim g$ exactly when “$f$ weakly pretends to be $g$”.

Our first theorem extends results of Tao [23] and Tao, Teräväinen [26] that correspond to the case $a(n)=n$ (see Section 2 for definitions of the notions used).

Theorem 1.1**.**

Let $a\colon{\mathbb{N}}\to{\mathbb{N}}$ be a deterministic and totally ergodic sequence. Let $\ell\in{\mathbb{N}}$ and $f_{1},\ldots,f_{\ell}\colon{\mathbb{N}}\to{\mathbb{U}}$ be multiplicative functions such that for every Dirichlet character $\chi$ we have $f_{1}\cdots f_{\ell}\nsim\chi$. Then for all $n_{1},\ldots,n_{\ell}\in{\mathbb{N}}$, we have

[TABLE]

Furthermore, the conclusion holds if $\ell=2$, $n_{1}\neq n_{2}$, and either $f_{1}$ or $f_{2}$ is strongly aperiodic.

Remarks**.**

$\bullet$ The conclusion is expected to hold for all deterministic sequences $a\colon{\mathbb{N}}\to{\mathbb{N}}$. But even relaxing the total ergodicity assumption to ergodicity seems difficult.

$\bullet$ Note that (2) is non-trivial even when $\ell=1$ (this case is also implicit in [8]).

Note that when $\ell$ is odd or $\ell=2$, the previous result applies to the case where all the multiplicative functions are equal to the Liouville or the Möbius function.

Henceforth, with $\mathbb{S}$ we denote the complex unit circle. Using the previous result for $\ell=2$ and $f_{1}=f$, $f_{2}=\bar{f}$, where $f\colon{\mathbb{N}}\to\mathbb{S}$ is a strongly aperiodic multiplicative function, we can immediately deduce using an argument from [24] (see [6, Proposition 2.3] for the needed result) the following:

Corollary 1.2**.**

Let $a\colon{\mathbb{N}}\to{\mathbb{N}}$ be a deterministic and totally ergodic sequence and $f\colon{\mathbb{N}}\to\mathbb{S}$ be a strongly aperiodic multiplicative function. Then

[TABLE]

Remark**.**

The same argument works if $f$ coincides with a strongly aperiodic multiplicative function $f\colon{\mathbb{N}}\to{\mathbb{U}}$ on a set with logarithmic density one and satisfies $\liminf_{N\to\infty}\mathbb{E}^{\log}_{n\in[N]}|f(n)|>0$.

When $a(n)=n$, the divergence in (3) was established by Tao [24] for every completely multiplicative function $f\colon{\mathbb{N}}\to\mathbb{S}$ and this was a decisive step in his solution of the Erdös discrepancy problem. It seems likely that the conclusion of the corollary also holds for every completely multiplicative function $f\colon{\mathbb{N}}\to\mathbb{S}$ but it is not clear how to prove this when $f$ is not strongly aperiodic.

1.3. Correlations along independent sequences

Next, we give results about correlations of multiplicative functions with shifts belonging to sets, or given by sequences, that satisfy certain linear independence properties.

Definition**.**

We say that a subset $S$ of ${\mathbb{N}}^{\ell}$ has independent elements if for every non-zero $k\in{\mathbb{Z}}^{\ell}$ the equation $k\cdot n=0$ has only finitely many solutions in $S$.

Remark**.**

The range of the three collections of sequences given in examples (i)-(iii) below form subsets of ${\mathbb{N}}^{\ell}$ with independent elements. These are “thin sets”, but there are also examples of subsets of ${\mathbb{N}}^{\ell}$ with independent elements that have density $1$; to see this, using a standard construction, take $Z$ to be a set of zero density that contains all but finitely many elements of each of the sets $\{n\in{\mathbb{N}}^{\ell}\colon k\cdot n=0\}$, where $k\in{\mathbb{Z}}^{\ell}$ is non-zero, and let $S:={\mathbb{N}}^{\ell}\setminus Z$.

Theorem 1.3**.**

Let $\ell\in{\mathbb{N}}$, $f_{0},\ldots,f_{\ell}\colon{\mathbb{N}}\to{\mathbb{U}}$ be multiplicative functions that admit log-correlations on ${\mathbf{M}}$, and suppose that at least one of them is strongly aperiodic. Furthermore, let $S$ be an infinite subset of ${\mathbb{N}}^{\ell}$ that has independent elements. We set $n_{0}:=0$ and denote the coordinates of $n\in{\mathbb{N}}^{\ell}$ with $n_{1},\ldots,n_{\ell}$. Then

[TABLE]

Remarks**.**

$\bullet$ Proving (4) under the weaker assumption that $S$ has distinct elements is as hard as Elliott’s conjecture (a generalization of the Chowla conjecture). In fact, if (4) holds for the set $S:=\{(n_{1}n,\ldots,n_{\ell}n),n\in{\mathbb{N}}\}$ for some specific $n_{1},\ldots,n_{\ell}\in{\mathbb{N}}$, then Theorem 4.3 below gives that $\mathbb{E}^{\log}_{m\in{\mathbf{M}}}\,\prod_{j=0}^{\ell}f_{j}(m+n_{j})=0$.

$\bullet$ Note that for $\ell$ odd, Theorem 1.3 is not covered by Theorem 1.1 (for $a(n)=n$) because the assumptions on the multiplicative functions in each result are different.

From Theorem 1.3 we can easily deduce (see Section 4) a result about correlations of multiplicative functions with shifts given by sequences that satisfy certain independence properties that we define next.

Definition**.**

Let $\ell\in{\mathbb{N}}$. We say that a collection of sequences $a_{1},\ldots,a_{\ell}\colon{\mathbb{N}}^{r}\to{\mathbb{N}}$

• •

independent if for every $k_{1},\ldots,k_{\ell}\in{\mathbb{Z}}$, not all of them zero, we have that $\sum_{j=1}^{\ell}k_{j}a_{j}(n)\neq 0$ for all but finitely many $n\in{\mathbb{N}}^{r}$.

• •

weakly independent if for every $k_{1},\ldots,k_{\ell}\in{\mathbb{Z}}$, not all of them zero, we have that $\sum_{j=1}^{\ell}k_{j}a_{j}(n)\neq 0$ outside a set of $n\in{\mathbb{N}}^{r}$ with density zero.

Restricting to the case $r=1$ we can easily verify that the following collections of sequences are independent:

1. (i)

$[n\alpha_{1}+\beta_{1}],\ldots,[n\alpha_{\ell}+\beta_{\ell}]$, where $\alpha_{1},\ldots,\alpha_{\ell}\in{\mathbb{R}}_{+}$ are rationally independent, meaning, all non-trivial integer combinations of the $\alpha_{1},\ldots,\alpha_{\ell}$ are non-zero, and $\beta_{1},\ldots,\beta_{\ell}\in{\mathbb{R}}$ are arbitrary. 2. (ii)

$a_{1}(n),\ldots,a_{\ell}(n)$, where $a_{1},\ldots,a_{\ell}\colon{\mathbb{N}}\to{\mathbb{N}}$ have different growth rates, meaning, they satisfy $\lim_{n\to\infty}a_{i}(n)/a_{j}(n)=0$ or $+\infty$ for $i\neq j$. 3. (iii)

$p_{1}(n),\ldots,p_{\ell}(n)$, where $p_{1},\ldots,p_{\ell}\colon{\mathbb{N}}\to{\mathbb{N}}$ are linearly independent polynomials.

If $\ell,r\in{\mathbb{N}}$ and $r\geq\ell$, it is easy to verify that any collection $L_{1},\ldots,L_{\ell}\colon{\mathbb{N}}^{r}\to{\mathbb{N}}$ of linearly independent linear forms is weakly independent, but if $r\geq 2$, then no such collection can be independent.

Theorem 1.4**.**

Let $\ell\in{\mathbb{N}}$, $f_{0},\ldots,f_{\ell}\colon{\mathbb{N}}\to{\mathbb{U}}$ be multiplicative functions that admit log-correlations on ${\mathbf{M}}$, and suppose that at least one of them is strongly aperiodic.

1. (i)

Let $a_{0}:=0$ and suppose that $a_{1},\ldots,a_{\ell}\colon{\mathbb{N}}^{r}\to{\mathbb{N}}$ are independent sequences. Then

[TABLE] 2. (ii)

Let $a_{0}:=0$ and suppose that $a_{1},\ldots,a_{\ell}\colon{\mathbb{N}}^{r}\to{\mathbb{N}}$ are weakly independent sequences. Then

[TABLE]

Lastly, we give a result regarding correlations of arbitrary multiplicative functions taking values on the (real) unit interval. For this, we need to impose an equidistribution assumption that we define next.

Definition**.**

Let $\ell,r\in{\mathbb{N}}$. We say that a collection of sequences $a_{1},\ldots,a_{\ell}\colon{\mathbb{N}}^{r}\to{\mathbb{N}}$ is jointly equidistributed in congruence classes if the sequence $(a_{1}(n),\ldots,a_{\ell}(n))_{n\in{\mathbb{N}}^{r}}$ is equidistributed in congruence classes. Equivalently, for all $u\in{\mathbb{N}}$ and all $k_{1},\ldots,k_{\ell}\in{\mathbb{Z}}$, not all of them multiples of $u$, we have ${\mathbb{E}}_{n\in{\mathbb{N}}^{r}}\,e^{\frac{2\pi i}{u}\,\sum_{j=1}^{\ell}k_{j}a_{j}(n)}=0$.

It is known that the collection of sequences $[n\alpha_{1}],\ldots,[n\alpha_{\ell}]$, where $1,\alpha_{1},\ldots,\alpha_{\ell}\in{\mathbb{R}}$ are rationally independent, and the collection $[n^{c_{1}}],\ldots,[n^{c_{\ell}}]$, where $c_{1},\ldots,c_{\ell}\in{\mathbb{R}}_{+}\setminus{\mathbb{N}}$ are distinct, are both jointly equidistributed in congruence classes. The same holds for any collection of linear forms $L_{1},\ldots,L_{\ell}\colon{\mathbb{N}}^{\ell}\to{\mathbb{N}}$ as long as the determinant of their coefficient matrix has absolute value $1$. As remarked before, the previous collections of sequences are also weakly independent.

Theorem 1.5**.**

Let $\ell,r\in{\mathbb{N}}$. Suppose that $a_{0}:=0$ and $a_{1},\ldots,a_{\ell}\colon{\mathbb{N}}^{r}\to{\mathbb{N}}$ are weakly independent sequences that are jointly equidistributed in congruence classes. Furthermore, suppose that the multiplicative functions $f_{0},\ldots,f_{\ell}\colon{\mathbb{N}}\to[-1,1]$ admit log-correlations on ${\mathbf{M}}$. Then

[TABLE]

Remarks**.**

$\bullet$ Identity (7) fails for complex valued multiplicative functions. For example, if $\ell=r=1$, $f_{0}(n):=n^{i}$, $f_{1}(n):=n^{-i}$, and $a_{1}(n):=n$, then $\mathbb{E}^{\log}_{n\in{\mathbb{N}}}\,f_{0}(n)=0$ but $\mathbb{E}^{\log}_{m\in{\mathbb{N}}}\,f_{0}(m)\,f_{1}(m+a_{1}(n))=1$ for every $n\in{\mathbb{N}}$.

$\bullet$ The multiplicativity assumption is used in an essential way, it is easy to verify that (7) is not always true if $f_{0},\ldots,f_{\ell}$ are allowed to be arbitrary sequences taking values in $[-1,1]$, even if these sequences are jointly totally ergodic.

$\bullet$ Examples of Dirichlet characters show that (7) fails if we replace the average ${\mathbb{E}}_{n\in{\mathbb{N}}^{r}}$ with the limit in density $\operatorname{D-lim}_{|n|\to\infty}$.

1.4. Sign patterns

Using the results of the previous subsections it is easy to deduce results on sign patterns attained by multiplicative functions. The next result is a consequence of Theorem 1.1 and extends [26, Corollary 1.10(i)], which corresponds to the case where $a(n)=n$ (the result in [26] is stated only for $f=\lambda$ but the argument given works in the more general setup of the next theorem).

Theorem 1.6**.**

Let $a\colon{\mathbb{N}}\to{\mathbb{N}}$ be a deterministic and totally ergodic sequence and $f\colon{\mathbb{N}}\to\{-1,1\}$ be a multiplicative function such that $f\nsim\chi$ for every Dirichlet character $\chi$ (it is known that then $f$ is also strongly aperiodic). Furthermore, let $n_{1},n_{2},\in{\mathbb{N}}$ be distinct, and $\epsilon_{0},\epsilon_{1},\epsilon_{2}\in\{-1,+1\}$. Then the logarithmic density of the set $\{m\in{\mathbb{N}}\colon f(a(m))=\epsilon_{0},\,f(a(m+n_{1}))=\epsilon_{1},\,f(a(m+n_{2}))=\epsilon_{2}\}$ is $\frac{1}{8}$.

Also, arguing as in [26, Corollary 7.2] we can deduce from Theorem 1.1 that if $n_{1},n_{2},n_{3}\in{\mathbb{N}}$ are distinct and $\epsilon_{0},\epsilon_{1},\epsilon_{2},\epsilon_{3}\in\{-1,+1\}$ are arbitrary, then the set

[TABLE]

has positive lower (natural) density.

In the next statement, if ${\mathbf{M}}$ is a sequence of intervals and $\Lambda\subset{\mathbb{N}}$, we define

[TABLE]

(note that we use logarithmic averages) assuming that the limit exists.

Theorem 1.7**.**

Let $\ell,r\in{\mathbb{N}}$. Let $f_{0},\ldots,f_{\ell}\colon{\mathbb{N}}\to\{-1,1\}$ be strongly aperiodic multiplicative functions that admit log-correlations on ${\mathbf{M}}$. Furthermore, if $n\in{\mathbb{N}}^{r}$ and $\epsilon_{0},\ldots,\epsilon_{\ell}\in\{-1,+1\}$, let ${\bf\epsilon}:=(\epsilon_{0},\ldots,\epsilon_{\ell})$ and

[TABLE]

1. (i)

If the sequences $a_{1},\ldots,a_{\ell}\colon{\mathbb{N}}^{r}\to{\mathbb{N}}$ are independent, then

[TABLE] 2. (ii)

If the sequences $a_{1},\ldots,a_{\ell}\colon{\mathbb{N}}^{r}\to{\mathbb{N}}$ are weakly independent, then

[TABLE]

Lastly, we state an immediate consequence of the previous result.

Corollary 1.8**.**

Let $\ell\in{\mathbb{N}}$. Suppose that $a_{1},\ldots,a_{\ell}\colon{\mathbb{N}}\to{\mathbb{N}}$ are sequences with different growth rates, or $a_{j}(n)=[n\alpha_{j}]$, $j=1,\ldots,\ell$, where $\alpha_{1},\ldots,\alpha_{\ell}\in{\mathbb{R}}$ are rationally independent. Then for all but finitely many $n\in{\mathbb{N}}$, for all $\epsilon_{0},\ldots,\epsilon_{\ell}\in\{-1,+1\}$, there exist (infinitely many) $m\in{\mathbb{N}}$ such that

[TABLE]

Furthermore, the same conclusion holds if in place of $\lambda$ we use any other strongly aperiodic multiplicative function $f\colon{\mathbb{N}}\to\{-1,1\}$.

We remark that a similar result is unknown for linearly dependent sequences, for example if $\ell=4$ and $a_{j}(n)=jn$ for $j=1,2,3,4$ (see [28] for related progress).

1.5. Proof strategy

To prove Theorem 1.1 we make essential use of a structural result from [7, 8] for measure preserving systems (called Furstenberg systems) naturally associated with arbitrary multiplicative functions $f_{1},\ldots,f_{\ell}\colon{\mathbb{N}}\to{\mathbb{U}}$. This structural result is used implicitly in the disjointness statement of Theorem 3.3 and gives the identities of Theorem 3.1. These identities allow us to deduce Theorem 1.1 from the main results in [23] and [26].

To prove Theorems 1.3 and 1.5 we first reinterpret them in ergodic terms using Proposition 2.1 and then use the identities of Theorem 4.3 in order to reduce matters to proving the ergodic theorems stated in Propositions 4.4 and 4.5 respectively. To prove these ergodic theorems we use the theory of characteristic factors (see [18, Chapter 21] for a description of the general method) and qualitative equidistribution results on nilmanifolds.

Lastly, we remark that the number theoretic results of Matomäki and Radziwiłł [21] and Matomäki, Radziwiłł, and Tao [22], on averages of multiplicative functions in short intervals, are used in an essential way in all our results except the first part of Theorem 1.1. In the ergodic setting, this number theoretic input translates to the fact that a certain function is orthogonal to the Kronecker factor of the system (see Part (i) of Proposition 4.2). Moreover, in the proof of Theorem 1.5 we use another fact from [21], that the mean value of a real valued bounded multiplicative function is essentially constant on the typical short interval, a property that fails for complex valued multiplicative functions.

1.6. Some open problems

In [7, 8] it was shown that any F-system of a collection of multiplicative functions $f_{1},\ldots,f_{\ell}\colon{\mathbb{N}}\to{\mathbb{U}}$ has no irrational spectrum. We are unable to prove a similar result for collections $f_{1}\circ a,\ldots,f_{\ell}\circ a\colon{\mathbb{N}}\to{\mathbb{U}}$, where $a\colon{\mathbb{N}}\to{\mathbb{N}}$ is a deterministic sequence even if it is totally ergodic. In fact, there is a serious obstacle in proving this even when $f_{1}=\cdots=f_{\ell}=\lambda$. The reason is that it is consistent with existing knowledge (though highly unlikely) that an F-system of $\lambda$ on some sequence of intervals ${\mathbf{M}}$ is isomorphic to the system defined by the transformation $T(x,y)=(x,y+x)$ acting on ${\mathbb{T}}^{2}$ with the Haar measure. If this is the case, then for $a(n)=[n\alpha]$, $n\in{\mathbb{N}}$, where $\alpha>1$ is irrational, we can check that $e^{2\pi i\alpha}$ belongs to the spectrum of the $F$-system of $\lambda\circ a$ on ${\mathbf{M}}$. A similar obstacle prevents us from proving a variant of Sarnak’s conjecture for $\lambda\circ a$ for ergodic weights, namely that

[TABLE]

whenever $a\colon{\mathbb{N}}\to{\mathbb{N}}$ is deterministic and totally ergodic and $w\colon{\mathbb{N}}\to{\mathbb{U}}$ is ergodic (in [7] this was established when $a(n)=n$).

In any case, the following statement seems plausible and if proved it would solve the problems just mentioned:

Problem 1**.**

Let $a\colon{\mathbb{N}}\to{\mathbb{N}}$ be a deterministic and totally ergodic sequence and for $\ell\in{\mathbb{N}}$ let $f_{1},\ldots,f_{\ell}\colon{\mathbb{N}}\to{\mathbb{U}}$ be multiplicative functions. Then all the joint correlations of the sequences $f_{1}\circ a,\ldots,f_{\ell}\circ a$ coincide with the joint correlations of $f_{1},\ldots,f_{\ell}$ (and thus the corresponding F-systems coincide).

Remark**.**

Equivalently, the conclusion asserts that

[TABLE]

for all $s\in{\mathbb{N}}$, $n_{1},\ldots,n_{s}\in{\mathbb{N}}$, and $g_{1},\ldots,g_{s}\in\{f_{1},\ldots,f_{\ell},\overline{f}_{1},\ldots,\overline{f}_{\ell}\}$, whenever both limits $\mathbb{E}^{\log}_{m\in{\mathbf{M}}}$ exist.

Lastly we mention a problem related both to Theorem 1.1 and Theorem 1.5.

Definition**.**

We say that a collection of sequences $a_{1},\ldots,a_{\ell}\colon{\mathbb{N}}\to{\mathbb{U}}$ is jointly totally ergodic if all its F-systems are totally ergodic.

Problem 2**.**

Let $\ell\in{\mathbb{N}}$. Suppose that the sequences $a_{1},\ldots,a_{\ell}\colon{\mathbb{N}}\to{\mathbb{N}}$ are deterministic, weakly independent, and jointly totally ergodic. If $f_{1},\ldots,f_{\ell}\colon{\mathbb{N}}\to[-1,1]$ are arbitrary multiplicative functions, then

[TABLE]

Remarks**.**

$\bullet$ A particular case of interest is when $f_{j}=\lambda$ and $a_{j}(n)=[n\alpha_{j}]$, $j=1,\ldots,\ell$, where $1,\alpha_{1},\ldots,\alpha_{\ell}$ are rationally independent real numbers.

$\bullet$ Identity (8) is false for complex valued multiplicative functions. For example if $\ell=2$, $f_{1}(n)=n^{i}$, $f_{2}(n)=n^{-i}$, and $a_{1}(n)=[n\alpha]$, $a_{2}(n)=[n\beta]$ with $1,\alpha,\beta$ rationally independent, then $\mathbb{E}^{\log}_{n\in{\mathbb{N}}}f_{1}(n)=0$ but $\mathbb{E}^{\log}_{n\in{\mathbb{N}}}\,f_{1}(a_{1}(n))\,f_{2}(a_{2}(n))=1$.

2. Background in ergodic theory and number theory

2.1. Measure preserving systems

Throughout the article, we make the standard assumption that all probability spaces $(X,{\mathcal{X}},\mu)$ considered are Lebesgue, meaning, $X$ can be given the structure of a compact metric space and ${\mathcal{X}}$ is its Borel $\sigma$-algebra. A measure preserving system, or simply a system, is a quadruple $(X,{\mathcal{X}},\mu,T)$ where $(X,{\mathcal{X}},\mu)$ is a probability space and $T\colon X\to X$ is an invertible, measurable, measure preserving transformation. We typically omit the $\sigma$-algebra ${\mathcal{X}}$ and write $(X,\mu,T)$. The system is ergodic if the only sets that are invariant by $T$ have measure [math] or $1$. It is totally ergodic if the system $(X,\mu,T^{n})$ is ergodic for every $n\in{\mathbb{N}}$. Throughout, for $n\in{\mathbb{N}}$ we denote by $T^{n}$ the composition $T\circ\cdots\circ T$ ($n$ times) and let $T^{-n}:=(T^{n})^{-1}$ and $T^{0}:=\operatorname{id}_{X}$. Also, for $F\in L^{1}(\mu)$ and $n\in{\mathbb{Z}}$ we denote by $T^{n}F$ the function $F\circ T^{n}$.

In order to avoid unnecessary repetition, we refer the reader to the article [7] for some other standard notions from ergodic theory. In particular, the reader will find in Section 2 and in Appendix A of [7] the definition of the terms factor, conditional expectation with respect to a factor, (rational) Kronecker factor, isomorphism, inverse limit, ergodic decomposition, joining, and disjoint systems; all these notions are used in this article.

2.2. Furstenberg systems associated with bounded sequences

For the purposes of this article, all averages in the definitions below are taken to be logarithmic. The reason is that we invoke results like Theorems 3.3 and 4.3 below that are only known when the joint Furstenberg systems are defined using logarithmic averages. This limitation comes from the number theoretic input used in their proofs, in particular the identities in [8, Theorem 3.1] that are based on the entropy decrement argument of Tao [23].

Definition**.**

Let ${\mathbf{M}}:=([M_{k}])_{k\in{\mathbb{N}}}$ be a sequence of intervals with $M_{k}\to\infty$. We say that a finite collection of bounded sequences $a_{1},\ldots,a_{\ell}\colon{\mathbb{N}}\to{\mathbb{U}}$ admits log-correlations on ${\mathbf{M}}$, if the limits

[TABLE]

exist for all $s\in{\mathbb{N}}$, all $n_{1},\ldots,n_{s}\in{\mathbb{Z}}$, and all $\tilde{a}_{1},\ldots,\tilde{a}_{s}\in\{a_{1},\ldots,a_{\ell},\overline{a}_{1},\ldots,\overline{a}_{\ell}\}$.

Remark**.**

Given $a_{1},\ldots,a_{\ell}\colon{\mathbb{Z}}\to\mathbb{U}$, using a diagonal argument, we get that every sequence of intervals ${\mathbf{M}}=([M_{k}])_{k\in{\mathbb{N}}}$ has a subsequence ${\mathbf{M}}^{\prime}=([M_{k}^{\prime}])_{k\in{\mathbb{N}}}$, such that the sequences $a_{1},\ldots,a_{\ell}$ admit log-correlations on ${\mathbf{M}}^{\prime}$.

For every finite collection of sequences that admits log-correlations on a given sequence of intervals, we use a variant of the correspondence principle of Furstenberg [11, 12] in order to associate a measure preserving system that captures the statistical properties of these sequences.

Definition**.**

Let $\ell\in{\mathbb{N}}$ and $a_{1},\ldots,a_{\ell}\colon{\mathbb{Z}}\to\mathbb{U}$ be sequences that admit log-correlations on ${\mathbf{M}}:=([M_{k}])_{k\in{\mathbb{N}}}$. We let $\mathcal{A}:=\{a_{1},\ldots,a_{\ell}\}$, $X:=(\mathbb{U}^{\ell})^{\mathbb{Z}}$, $T$ be the shift transformation on $X$, and $\mu$ be the weak-star limit of the sequence $(\mathbb{E}^{\log}_{m\in[M_{k}]}\,\delta_{T^{m}a})_{k\in{\mathbb{N}}}$ where $a:=(a_{1},\ldots,a_{\ell})$ is thought of as an element of $X$. We call $(X,\mu,T)$ the joint Furstenberg system associated with $\mathcal{A}$ on ${\mathbf{M}}$, or simply, the F-system of $\mathcal{A}$ on ${\mathbf{M}}$.

Remark**.**

If we are given sequences $a_{1},\ldots,a_{\ell}\colon{\mathbb{N}}\to\mathbb{U}$, we extend them to ${\mathbb{Z}}$ in an arbitrary way; then the measure $\mu$ will not depend on the extension.

We state explicitly some useful identities that are implicit in the previous definition.

Proposition 2.1**.**

Let $\ell\in{\mathbb{N}}$ and $a_{1},\ldots,a_{\ell}\colon{\mathbb{Z}}\to\mathbb{U}$ be sequences that admit log-correlations on ${\mathbf{M}}:=([M_{k}])_{k\in{\mathbb{N}}}$ and let $(X,\mu,T)$ be the corresponding F-system on ${\mathbf{M}}$. For $j=1,\ldots,\ell$, consider the functions $F_{j}\in C(X)$ defined by $F_{j}(x):=x_{j}(0)$, where we assume that $x\in X$ has the form $(x_{1}(n),\ldots,x_{\ell}(n))_{n\in{\mathbb{Z}}}$. Then

[TABLE]

*for all $n_{1},\ldots,n_{\ell}\in{\mathbb{Z}}$, *

Note that a collection of sequences $a_{1},\ldots,a_{\ell}\colon{\mathbb{Z}}\to\mathbb{U}$ may have several non-isomorphic F-systems depending on which sequence of intervals ${\mathbf{M}}$ we use in the evaluation of their joint correlations. We call any such system an F-system of $a_{1},\ldots,a_{\ell}$.

2.3. Multiplicative functions

We denote by $\mathcal{M}$ the set of all multiplicative functions $f\colon{\mathbb{N}}\to\mathbb{U}$, where $\mathbb{U}$ is the complex unit disc.

We make use of the following notion introduced in [22]:

Definition**.**

Let ${\mathbb{D}}\colon\mathcal{M}\times\mathcal{M}\times\mathbb{R}_{+}\to[0,\infty]$ be given by

[TABLE]

and $M\colon\mathcal{M}\times\mathbb{N}\to[0,\infty)$ be given by

[TABLE]

The multiplicative function $f\in\mathcal{M}$ is strongly aperiodic (or strongly non-pretentious using terminology from [13]) if $M(f\cdot\chi;N)\to\infty$ as $N\to\infty$ for every Dirichlet character $\chi$.

It is known that the Möbius and the Liouville functions are strongly aperiodic (see [23]). More generally, if $f(p)$ is a nontrivial $d$-th root of unity for all $p\in{\mathbb{P}}$, then $f$ is strongly aperiodic (see for example [5, Corollary 6.2]).

The hypothesis of strong aperiodicity is useful for our purposes because it gives us access to the ergodic property stated in Part (i) of Proposition 4.2 below and also to the following result of Tao [23, Corollary 1.5]:

Theorem 2.2**.**

If $f,g\colon{\mathbb{N}}\to{\mathbb{U}}$ are multiplicative functions and at least one of them is strongly aperiodic, then

[TABLE]

for every $n\in{\mathbb{N}}$.

3. Proof of results about deterministic sequences

Theorem 1.1 is a consequence of the main results in [23] and [26] and the following correlation identities:

Theorem 3.1**.**

Let $a\colon{\mathbb{N}}\to{\mathbb{N}}$ be a deterministic and totally ergodic sequence and for $\ell\in{\mathbb{N}}$ let $f_{1},\ldots,f_{\ell}\colon{\mathbb{N}}\to{\mathbb{U}}$ be multiplicative functions. Then every sequence of intervals ${\mathbf{M}}$ has a subsequence ${\mathbf{M}}^{\prime}$ such that for every $n_{1},\ldots,n_{\ell}\in{\mathbb{N}}$ all limits below exist and we have the identity

[TABLE]

Remark**.**

The identity fails if we do not assume that $f_{1},\ldots,f_{\ell}$ are multiplicative. It also fails if we remove the assumption that $a\colon{\mathbb{N}}\to{\mathbb{N}}$ is deterministic, or if we replace the total ergodicity assumption with ergodicity.

Proof of Theorem 1.1 assuming Theorem 3.1.

Let $f_{1},\ldots,f_{\ell}$ be multiplicative functions that satisfy the assumptions of Theorem 1.1. Arguing by contradiction, suppose that (2) fails. Then there exist a sequence of intervals ${\mathbf{M}}$ and $n_{1},\ldots,n_{\ell}\in{\mathbb{N}}$ such that

[TABLE]

By Theorem 3.1 the sequence of intervals ${\mathbf{M}}$ has a subsequence ${\mathbf{M}}^{\prime}$ such that all limits below exist and we have

[TABLE]

Using the main result in [26] if $f_{1}\cdots f_{\ell}\nsim\chi$ for every Dirichlet character $\chi$, and the main result in [23] (see Theorem 2.2 in this article) if $\ell=2$, $n_{1}\neq n_{2}$ (note that then $a(n+n_{1})\neq a(n+n_{2})$), and either $f_{1}$ or $f_{2}$ is strongly aperiodic, we get that

[TABLE]

for every $n\in{\mathbb{N}}$. This contradicts (10) and completes the proof. ∎

So our goal is to prove Theorem 3.1.

3.1. Setup

We prove Theorem 3.1 via a disjointness argument. Our initial setup loosely follows the one in [19, Lemma 4.1]. With ${\mathbb{Z}}_{+}$ we denote the set of non-negative integers. First, we establish a correspondence between strictly increasing sequences $a\colon{\mathbb{Z}}_{+}\to{\mathbb{Z}}_{+}$ and elements of the sequence space $Y:=\{0,1\}^{{\mathbb{Z}}_{+}}$. Let

[TABLE]

where ${\bf 0}$ denotes the element of $Y$ that has all its coordinates [math]. For $y\in Y$ we let $\tau_{y}\colon{\mathbb{Z}}_{+}\to{\mathbb{Z}}_{+}$ be defined by

[TABLE]

if $y\not\in Z$ and $\tau_{y}(n)=0$ if $y\in Z$. If $a\colon{\mathbb{Z}}_{+}\to{\mathbb{Z}}_{+}$ is a strictly increasing sequence with range $A$, then it defines the point $y_{a}:={\bf 1}_{A}\in Y$. On the other hand, we have $\tau_{y_{a}}(j)=a(j)$ for every $j\in{\mathbb{Z}}_{+}$, so the map $y\mapsto\tau_{y}$ sends the point $y_{a}\in\{0,1\}^{{\mathbb{Z}}_{+}}$ (note that $y_{a}\notin Z$) to the sequence $a\colon{\mathbb{Z}}_{+}\to{\mathbb{Z}}_{+}$.

Next, given sequences $b_{1},\ldots,b_{\ell}\colon{\mathbb{Z}}\to{\mathbb{U}}$ and $a\colon{\mathbb{Z}}_{+}\to{\mathbb{Z}}_{+}$ we want to reinterpret correlations of the sequences $b_{1}\circ a,\ldots,b_{\ell}\circ a$ in dynamical terms. Let $X:=({\mathbb{U}}^{\ell})^{\mathbb{Z}}$ where we think of elements $x$ of $X$ as $\ell$-tuples $(x_{1},\ldots,x_{\ell})$ with $x_{1},\ldots,x_{\ell}\in{\mathbb{U}}^{\mathbb{Z}}$. Let $R,S$ be the shifts on the spaces $X,Y$ correspondingly. For $n\in{\mathbb{Z}}_{+}$ and $j=1,\ldots,\ell$, we define the function $F_{j,n}\colon X\times Y\to{\mathbb{U}}$ by

[TABLE]

Note that for every $n\in{\mathbb{Z}}_{+}$ the function $y\mapsto\tau_{y}(n)$ is continuous on $Y\setminus Z$. Using this it is easy to verify that for every $n\in{\mathbb{Z}}_{+}$ and $j\in\{1,\ldots,\ell\}$ we have

[TABLE]

and

[TABLE]

Let $y\notin Z$. An easy computation shows that for every $m\in{\mathbb{N}}$ and $n\in{\mathbb{Z}}_{+}$ we have

[TABLE]

Combining the last two identities we get for every $j\in\{1,\ldots,\ell\}$ and every $m\in{\mathbb{N}}$ and $n\in{\mathbb{Z}}_{+}$ that

[TABLE]

Hence, for every $m\in{\mathbb{N}}$ and $n_{1},\ldots,n_{\ell}\in{\mathbb{Z}}_{+}$ we have that

[TABLE]

Therefore, if we let $k_{y}(0):=0$ and

[TABLE]

we have for every $M\in{\mathbb{N}}$ and $c(0),\ldots,c(k_{y}(M))\in{\mathbb{C}}$ that

[TABLE]

Let now $b_{1},\ldots,b_{\ell}\colon{\mathbb{Z}}\to{\mathbb{U}}$ be arbitrary sequences and $a\colon{\mathbb{Z}}_{+}\to{\mathbb{Z}}_{+}$ be a strictly increasing sequence with range a set of density $\alpha>0$, or equivalently,

[TABLE]

Clearly $y_{a}\notin Z$. We let $b:=(b_{1},\ldots,b_{\ell})\in X$. Using that $\tau_{y_{a}}(j)=a(j)$, $j\in{\mathbb{Z}}_{+}$, equation (16), and the scale invariance of logarithmic averages, we deduce using (15) for $c(m):=\frac{1}{m}$ if $m\in{\mathbb{N}}$ and $c(0):=0$, that if for some sequence of intervals ${\mathbf{M}}$ the limit $\mathbb{E}^{\log}_{m\in{\mathbf{M}}}$ on the left hand side below exists, then the same holds for the limit on the right hand side and we have the identity

[TABLE]

This completes the needed dynamical reinterpretation of the correlations of the sequences $b_{1}\circ a,\ldots,b_{\ell}\circ a$ that will be used shortly.

3.2. Proof of Theorem 3.1

The first ingredient in the proof of Theorem 3.1 is the following result:

Proposition 3.2**.**

Let $a\colon{\mathbb{N}}\to{\mathbb{N}}$ be a strictly increasing sequence with range a set $A$ of positive density. Let $\ell\in{\mathbb{N}}$ and suppose that all the F-systems of the sequences $b_{1},\ldots,b_{\ell}\colon{\mathbb{N}}\to{\mathbb{U}}$ are disjoint from all the F-systems of ${\bf 1}_{A}$. Then every sequence of intervals ${\mathbf{M}}$ has a subsequence ${\mathbf{M}}^{\prime}$ such that for every $n_{1},\ldots,n_{\ell}\in{\mathbb{N}}$ all limits below exist and we have the identity

[TABLE]

Proof.

We follow the notation established in Section 3.1. Using a diagonal argument we can find a subsequence ${\mathbf{M}}^{\prime}=([M_{k}^{\prime}])_{k\in{\mathbb{N}}}$ of ${\mathbf{M}}$, such that for every $n_{1},\ldots,n_{\ell}\in{\mathbb{N}}$ all limits below exist and by (17) we have the identity

[TABLE]

where $y_{a}:={\bf 1}_{A}$, $\alpha>0$ is the density of the set $A$, and

[TABLE]

By passing to a subsequence of ${\mathbf{M}}^{\prime}$, which we denote again by ${\mathbf{M}}^{\prime}$, we can assume that the sequence of measures $(\mathbb{E}^{\log}_{m\in[M_{k}^{\prime}]}\,\delta_{(R^{m}b,S^{m}y_{a})})_{k\in{\mathbb{N}}}$ converges weak-star to a probability measure $\rho$ on $X\times Y$. Let $\mu$ and $\nu$ be the marginals of $\rho$ and $R,S$ be the shift transformations on $X$, $Y$ respectively. Then $\rho$ is $R\times S$-invariant and

[TABLE]

where all implicit limits are weak-star limits. Then $(X,\mu,R)$ is an $F$-system of the sequences $b_{1},\ldots,b_{\ell}$ and $(Y,\nu,S)$ is an F-system of $y_{a}={\bf 1}_{A}$. By assumption, the two systems are disjoint, hence

[TABLE]

Next, we claim that the set $Z^{*}$, defined in (11), satisfies $\nu(Z^{*})=0$. Indeed, note that for $y\in Z^{*}$ the sets $T^{-n}\{y\}$, $n\in{\mathbb{N}}$, are disjoint. Using the shift invariance of the probability measure $\nu$ we deduce that $\nu(\{y\})=0$ for every $y\in Z^{*}$, and since $Z^{*}$ is a countable set, we conclude that $\nu(Z^{*})=0$.

Since $\nu(Z^{*})=0$ and by (12) the function $F$ is continuous on $X\times(Y\setminus Z^{*})$ and $\rho$ is a joining of $\mu$ and $\nu$, the function $F$ is continuous $\rho$-almost everywhere. Hence,

[TABLE]

where the last identity follows from (20) and Fubini’s theorem. Since by (12) the function $F$ is continuous on $X\times(Y\setminus Z^{*})$, we get using the bounded convergence theorem that the function $G\colon Y\to{\mathbb{C}}$ defined by $G(y):=\int F(x,y)\,d\mu(x)$, $y\in Y$, is continuous on $Y\setminus Z^{*}$. Since $\nu(Z^{*})=0$, the function $G$ is continuous $\nu$-almost everywhere, so we have

[TABLE]

Moreover, by (13), for every fixed $y\in Y$ the function $x\mapsto F(x,y)$ is continuous on $X$, hence for every $n\in{\mathbb{N}}$ we have

[TABLE]

Combining the last three identities we get

[TABLE]

Using the definition of the function $F$ in (19), identity (14), and the fact that $\tau_{y_{a}}(j)=a(j)$, $j\in{\mathbb{Z}}_{+}$, we get for every $m,n\in{\mathbb{N}}$ that

[TABLE]

Using (22) and arguing exactly as in the last part of Section 3.1 we deduce that

[TABLE]

Combining (18), (21), (23) (and using that $\alpha\neq 0$), we get the asserted identity. ∎

The next result is a crucial element in the proof of Theorem 1.1 and follows by combining the structural result of [8, Theorem 1.5] with the disjointness statement of [7, Proposition 3.12].

Theorem 3.3**.**

All F-systems of any bounded collection of multiplicative functions with values on the complex unit disc are disjoint from all zero entropy totally ergodic systems.

Combining the previous two results we can now prove Theorem 3.1.

Proof of Theorem 3.1.

By assumption, the sequence $a\colon{\mathbb{N}}\to{\mathbb{N}}$ is strictly increasing and its range, which we denote by $A$, has positive density. Also by assumption, all F-systems of ${\bf 1}_{A}$ have zero entropy and are totally ergodic. It follows from Theorem 3.3 that all F-systems of the collection $f_{1},\ldots,f_{\ell}\colon{\mathbb{N}}\to{\mathbb{U}}$ are disjoint from all F-systems of ${\bf 1}_{A}$. Hence, Proposition 3.2 applies and gives that the conclusion of Theorem 3.1 holds. ∎

4. Proof of results about independent sequences

In this section we prove Theorems 1.3, 1.4, 1.5.

We start with a reduction, we show that Theorem 1.4 follows from the following more general result:

Theorem 4.1**.**

Let $\ell\in{\mathbb{N}}$. Let $f_{0},\ldots,f_{\ell}\colon{\mathbb{N}}\to{\mathbb{U}}$ be multiplicative functions that admit log-correlations on ${\mathbf{M}}$ and suppose that at least one of them is strongly aperiodic. Furthermore, let $a_{0}:=0$ and $a_{1},\ldots,a_{\ell}\colon{\mathbb{N}}^{r}\to{\mathbb{N}}$ be sequences, and $R$ be an infinite subset of ${\mathbb{N}}^{r}$ such that the set $S:=\{(a_{1}(n),\ldots,a_{\ell}(n))\colon n\in R\}$ has independent elements (see definition in Section 1.3). Then

[TABLE]

Proof of Theorem 1.4 assuming Theorem 4.1..

Part $(i)$ of Theorem 1.4 follows at once, since for independent sequences $a_{1},\ldots,a_{\ell}$, for $R:={\mathbb{N}}^{r}$, the set $S$ in the statement of Theorem 4.1 has independent elements. Hence, (24) holds with $R={\mathbb{N}}^{r}$, so (5) holds.

We prove Part $(ii)$ of Theorem 1.4. Since the sequences $a_{1},\ldots,a_{\ell}$ are weakly independent all the sets $Z_{k_{1},\ldots,k_{\ell}}:=\{n\in{\mathbb{N}}^{r}\colon k_{1}a_{1}(n)+\cdots+k_{\ell}a_{\ell}(n)=0\}$, $k_{1},\ldots,k_{\ell}\in{\mathbb{Z}}$, have zero density unless $k_{1}=\cdots=k_{\ell}=0$. Since the collection of all these sets is countable, it is well known that there exists a subset $Z$ of ${\mathbb{N}}^{r}$ that has zero density and such that $Z_{k_{1},\ldots,k_{\ell}}\setminus Z$ is finite for all $k_{1},\ldots,k_{\ell}\in{\mathbb{Z}}$ not all of them [math]. Then for $R:={\mathbb{N}}^{r}\setminus Z$ the set $S$ in the statement of Theorem 4.1 has independent elements. Hence, (24) holds for this set $R$, and since the complement of $R$ has density zero, we get (6). ∎

Next we show that Theorem 4.1 follows from Theorem 1.3.

Proof of Theorem 4.1 assuming Theorem 1.3..

If $\ell=1$, then the result follows from Theorem 2.2. So we can assume that $\ell\geq 2$. In this case we have that $\lim_{n\to\infty,n\in R}|(a_{1}(n),\ldots,a_{\ell}(n))|=\infty$, because otherwise for $n\in R$ the vectors $(a_{1}(n),\ldots,a_{\ell}(n))$ attain some fixed value infinitely often, and this easily contradicts our assumption that the set $S$ in the statement of Theorem 4.1 has independent elements. Using this, we deduce (24) from (4). ∎

Hence, it remains to prove Theorems 1.3 and 1.5.

4.1. Ergodic feedback from number theory

We start by translating some input from number theory to useful ergodic properties.

Proposition 4.2**.**

Let $\ell\in{\mathbb{N}}$. Let $(X,\mu,T)$ be an F-system on ${\mathbf{M}}$ of the multiplicative functions $f_{1},\ldots,f_{\ell}\colon{\mathbb{N}}\to{\mathbb{U}}$ and $F_{1},\ldots,F_{\ell}$ be the functions of Proposition 2.1.

1. (i)

If $f_{1}$ is strongly aperiodic, then the function $F_{1}$ is orthogonal to the Kronecker factor of the system, meaning, it is orthogonal to all eigenfunctions of the system. 2. (ii)

If $f_{1}$ is real valued, then the function $F_{1}$ satisfies ${\mathbb{E}}(F_{1}|\mathcal{I})=\int F_{1}\,d\mu$, where $\mathcal{I}:=\{A\in{\mathcal{X}}\colon T^{-1}A=A\}$.

Remark**.**

Part (ii) fails for complex valued multiplicative functions. For example, if $f_{1}(n):=n^{i}$, $n\in{\mathbb{N}}$, then it can be shown that ${\mathbb{E}}(F_{1}|\mathcal{I})=F_{1}$ but $\int F_{1}\,d\mu=0$.

Proof.

For (i) the number theoretic input needed is [22, Theorem B.1] and the deduction can be found in the proof of [5, Proposition 5.1].

For (ii) the number theoretic input needed is [21, Theorem 1]. It implies that

[TABLE]

where $\alpha:={\mathbb{E}}_{n\in{\mathbb{N}}}f_{1}(n)$ (the limit is known to exist by a result of Wirsing [31]). This implies that

[TABLE]

where one sees that the limits $\mathbb{E}^{\log}_{m\in{\mathbf{M}}}$ exist by expanding the square and using our assumption that $f_{1}$ admits log-correlations on ${\mathbf{M}}$. Expanding the square and using Proposition 2.1 we get that

[TABLE]

Using the mean ergodic theorem we deduce that

[TABLE]

Hence, ${\mathbb{E}}(F_{1}|\mathcal{I})=\alpha=\int F_{1}\,d\mu$. ∎

4.2. Reduction to ergodic statements

In this subsection we show that Theorems 1.3 and 1.5 follow from two ergodic statements that we prove subsequently. In order to carry out the needed reduction we will make crucial use of certain identities satisfied by F-systems of multiplicative functions. They are based on work in [23] and [26] and are proved in [8, Theorem 3.8]. Henceforth, for $d\in{\mathbb{N}}$ we let ${\mathbb{P}}_{d}:={\mathbb{P}}\cap(d{\mathbb{Z}}+1)$.

Theorem 4.3**.**

Let $\ell\in{\mathbb{N}}$ and $f_{1},\ldots,f_{\ell}\colon{\mathbb{Z}}\to\mathbb{U}$ be multiplicative functions. There exists $d\in{\mathbb{N}}$ such that the following holds: If $(X,\mu,T)$ is an F-system of $f_{1},\ldots,f_{\ell}$ and if $F_{1},\ldots,F_{\ell}$ are as in Proposition 2.1, then we have

[TABLE]

for all $n_{1},\ldots,n_{\ell}\in{\mathbb{Z}}$.

We will show that Theorem 1.3 follows from the following result:

Proposition 4.4**.**

Let $\ell\in{\mathbb{N}}$ and $S$ be a subset of ${\mathbb{N}}^{\ell}$ with independent elements. Suppose that $(X,\mu,T)$ is a system and $F_{0},\ldots,F_{\ell}\in L^{\infty}(\mu)$ are functions at least one of which is orthogonal to the rational Kronecker factor of the system. We set $n_{0}:=0$ and denote the coordinates of $n\in{\mathbb{N}}^{\ell}$ by $n_{1},\ldots,n_{\ell}$. Then for every $d\in{\mathbb{N}}$ all limits ${\mathbb{E}}_{p\in{\mathbb{P}}_{d}}$ below exist and we have

[TABLE]

Remark**.**

A function is orthogonal to the rational Kronecker factor of a system $(X,\mu,T)$ if it is orthogonal to any function $F\in L^{\infty}(\mu)$ that satisfies $TF=e^{2\pi i\alpha}\,F$ for some $\alpha\in\mathbb{Q}$.

We will show that Theorem 1.5 follows from the following result:

Proposition 4.5**.**

Let $\ell,r\in{\mathbb{N}}$. Let also $a_{0}:=0$ and suppose that $a_{1},\ldots,a_{\ell}\colon{\mathbb{N}}^{r}\to{\mathbb{N}}$ are weakly independent sequences that are jointly equidistributed in congruence classes. Then for every $d\in{\mathbb{N}}$, for all ergodic systems $(X,\mu,T)$ and functions $F_{0},\ldots,F_{\ell}\in L^{\infty}(\mu)$, all the limits below exist and we have

[TABLE]

Remark**.**

Examples of periodic systems show that (27) fails if we replace ${\mathbb{E}}_{n\in{\mathbb{N}}^{r}}$ with $\operatorname{D-lim}_{|n|\to\infty}$.

Proof of Theorems 1.3 and 1.5 assuming Propositions 4.4 and 4.5.

First we prove Theorem 1.3 assuming Proposition 4.4. Let $(X,\mu,T)$ be the $F$-system of $f_{0},\ldots,f_{\ell}$ on ${\mathbf{M}}$. By Theorem 4.3 there exist $d\in{\mathbb{N}}$ and functions $F_{0},\ldots,F_{\ell}\in L^{\infty}(\mu)$ such that for all $n_{0},\ldots,n_{\ell}\in{\mathbb{Z}}$ we have

[TABLE]

Suppose that $f_{j_{0}}$ is strongly aperiodic for some $j_{0}\in\{0,\ldots,\ell\}$. Then by Part (i) of Proposition 4.2 the function $F_{j_{0}}$ is orthogonal to the Kronecker factor of the system $(X,\mu,T)$. Using this and identity (28), we deduce from Proposition 4.4 that Theorem 1.3 holds.

Next we prove Theorem 1.5 assuming Proposition 4.5. Let $\mu=\int\mu_{x}\,d\mu$ be the ergodic decomposition of the measure $\mu$. We have that

[TABLE]

where we used the bounded convergence theorem twice, the first time we used that the limits ${\mathbb{E}}_{p\in{\mathbb{P}}_{d}}\int\prod_{j=0}^{\ell}T^{pa_{j}(n)}F_{j}\,d\mu_{x}$ exist for every $n\in{\mathbb{N}}$ (see Theorem 4.7 below) and the second time we used that the limits {\mathbb{E}}_{n\in{\mathbb{N}}^{r}}\big{(}{\mathbb{E}}_{p\in{\mathbb{P}}_{d}}\int\prod_{j=0}^{\ell}T^{pa_{j}(n)}F_{j}\,d\mu_{x}\big{)} exist by Proposition 4.5. Using Proposition 4.5 once more, we get that

[TABLE]

where we used that for every $F\in L^{\infty}(\mu)$ for $\mu$ almost every $x\in X$ we have that ${\mathbb{E}}(F|\mathcal{I})(x)=\int F\,d\mu_{x}$ (see for example [18, Page 37]).

By Part (ii) of Proposition 4.2 we have that ${\mathbb{E}}(F_{j}|\mathcal{I})=\int F_{j}\,d\mu$ for $j=0,\ldots,\ell$ (here we made crucial use of the fact that the multiplicative functions are real valued). Combining the above with identity (28) and the fact that (by (9)) $\int F_{j}\,d\mu=\mathbb{E}^{\log}_{m\in{\mathbf{M}}}\,f_{j}(m)$ for $j=1,\ldots,\ell$, we get that Theorem 1.5 holds. ∎

Thus, it remains to prove the ergodic statements of Propositions 4.4 and 4.5. We do this in the remaining subsections.

4.3. Nilsystems, nilcharacters, and nilfactors

If $G$ is a group we let $G_{1}:=G$ and $G_{j+1}:=[G,G_{j}]$, $j\in\mathbb{N}$. We say that $G$ is $s$-step nilpotent if $G_{s+1}$ is the trivial group. An $s$-step nilmanifold is a homogeneous space $X=G/\Gamma$, where $G$ is an $s$-step nilpotent Lie group and $\Gamma$ is a discrete cocompact subgroup of $G$. With $e_{X}$ we denote the image in $X$ of the unit element of $G$. An $s$-step nilsystem is a system of the form $(X,{\mathcal{X}},m_{X},T_{b})$, where $X=G/\Gamma$ is an $s$-step nilmanifold, $b\in G$, $T_{b}\colon X\to X$ is defined by $T_{b}(g\cdot e_{X})\mathrel{\mathop{:}}=(bg)\cdot e_{X}$ for $g\in G$, $m_{X}$ is the normalized Haar measure on $X$, and ${\mathcal{X}}$ is the completion of the Borel $\sigma$-algebra of $G/\Gamma$. We call the map $T_{b}$ or the element $b$ a nilrotation. If $T_{b}$ acts ergodically we call $b$ an ergodic nilrotation.

With $G^{0}$ we denote the connected component of the identity element in $G$. When we are working with an ergodic nilsystem, we can assume that the space $X$ is represented as $X=G/\Gamma$ where $G=\langle G^{0},b\rangle$ and $\Gamma$ does not contains any non-trivial normal subgroups of $G$ (see pages 100 and 177 in [18] or [2, Section 4.1]). Henceforth, we are going to use these properties without further reference. When we work with such a representation we have that for $j\geq 2$ the commutator subgroups $G_{j}$ are connected (see [18, Page 155] or [2, Theorem 4.1]). Hence, in an $s$-step nilmanifold with $s\geq 2$, the subgroup $G_{s}$ is connected and the Abelian group $K_{s}:=G_{s}/(G_{s}\cap\Gamma)$ is a finite dimensional torus (perhaps the trivial one). Let $\widehat{K_{s}}$ be the dual group of $K_{s}$; it consists of the characters of $G_{s}$ that are $(\Gamma\cap G_{s})$-invariant. An $s$-step nilcharacter of $X$ (often called a vertical nilcharacter) with frequency $\chi$, where $\chi\in\widehat{K_{s}}$, is a function $\Phi\in{\mathbb{C}}(X)$ that satisfies

[TABLE]

If $\chi$ is a non-trivial character of $K_{s}$, we say that $\Phi$ is a non-trivial $s$-step nilcharacter, otherwise we say that it is a trivial $s$-step nilcharacter. It follows from (29) that every non-trivial $s$-step nilcharacter has zero integral. It is also known that the linear span of $s$-step nilcharacters is dense in $C(X)$ with the uniform norm (see for example [15, Proof of Lemma 2.7]).

If the nilmanifold $X$ is not connected, let $X^{0}$ be the connected component of $e_{X}$ in $X$. Then for $s\geq 2$ the restriction of a non-trivial $s$-step nilcharacter $\Phi$ of $X$ onto $X^{0}$ is a non-trivial $s$-step nilcharacter of $X^{0}$ with the same frequency (see [4, Section 3.3]).

Let $(X,\mu,T)$ be an ergodic system and for $k\in{\mathbb{N}}$ let $(Z_{k},{\mathcal{Z}}_{k},\mu_{k},T)$ be the factor of order $k$ of $X$ as defined in [18, Chapter 9] (we abuse notation and denote the transformation on $Z_{k}$ by $T$). The following result was proved in [17]:

Theorem 4.6**.**

If $(X,\mu,T)$ is an ergodic system, then for every $k\in{\mathbb{N}}$ the system $(Z_{k},{\mathcal{Z}}_{k},\mu_{k},T)$ is an inverse limit of ergodic $k$-step nilsystems.

We remark that properties of inverse limits imply that if $(X,\mu,T)$ is an ergodic system, then for every $F\in L^{\infty}(Z_{k},\mu_{k})$ and every $\varepsilon>0$, there exist a $k$-step nilsystem $(X^{\prime},m_{X^{\prime}},T^{\prime})$, a factor map $\pi\colon Z_{k}\to X^{\prime}$, and a function $F^{\prime}\in L^{\infty}(m_{X^{\prime}})$ (in fact, we can take $F^{\prime}:={\mathbb{E}}(F|X^{\prime})$), such that $\left\|F-F^{\prime}\circ\pi\right\|_{L^{1}(\mu)}\leq\varepsilon$. Furthermore, the function $F^{\prime}$ can be chosen so that if $F$ is orthogonal to the (rational) Kronecker factor of the system $(Z_{k},{\mathcal{Z}}_{k},\mu_{k},T)$, then $F^{\prime}$ is orthogonal to the (rational) Kronecker factor of the system $(X^{\prime},m_{X^{\prime}},T^{\prime})$.

4.4. Reduction to statements about nilsystems

To carry out our reductions further we will use the following convergence result:

Theorem 4.7**.**

Let $(X,\mu,T)$ be a system and $d,\ell\in{\mathbb{N}}$. Then for every $F_{1},\ldots,F_{\ell}\in L^{\infty}(\mu)$, the following limit exists in $L^{2}(\mu)$

[TABLE]

Furthermore, the factor $Z_{\ell}$ (defined in Section 4.3) is characteristic for mean convergence of these averages, meaning, if ${\mathbb{E}}(F_{j}|Z_{\ell})=0$ for some $j\in\{1,\ldots,\ell\}$, then the averages converge to [math] in $L^{2}(\mu)$.

Remark**.**

We note that for $\ell\geq 2$ the smaller factor $Z_{\ell-1}$ is characteristic for convergence of the previous averages, but we will not need this.

We remark that for $d=1$ the convergence part of this result follows from [32] and the part about characteristic factors from [9] (conditional to some conjectures obtained later in [14, 16]). The statement for general $d\in{\mathbb{N}}$ follows by using the $d=1$ case for product systems of the form $T\times R$ acting on $X\times{\mathbb{Z}}/(d{\mathbb{Z}})$ with the product measure, where $R$ is the shift on ${\mathbb{Z}}/(d{\mathbb{Z}})$, and for the functions $F_{1}\otimes{\bf 1}_{d{\mathbb{Z}}+1},F_{2},\ldots,F_{\ell}$. For the statement on characteristic factors one also uses the fact that for every $\ell\in{\mathbb{N}}$ if ${\mathbb{E}}(F|Z_{\ell}(T))=0$, then also ${\mathbb{E}}(F|Z_{\ell}(T\times R))=0$ and ${\mathbb{E}}(F\otimes{\bf 1}_{d{\mathbb{Z}}+1}|Z_{\ell}(T\times R))=0$.

Proposition 4.8**.**

If Proposition 4.4 holds for every ergodic nilsystem, then it holds for every system.

Proof.

First we use an ergodic decomposition argument to show that it suffices to verify the statement for ergodic systems. The proof of this reduction is the same as the one used in the proof of Theorem 1.5 assuming Proposition 4.5. The only additional ingredient needed is the well known fact that if $\mu=\int\mu_{x}\,d\mu$ is the ergodic decomposition of the measure $\mu$ and a function $F$ is orthogonal to the rational Kronecker factor of the system $(X,\mu,T)$, then for $\mu$ almost every $x\in X$ the function $F$ is orthogonal to the rational Kronecker factor of the system $(X,\mu_{x},T)$.

Next, note that since the set $S$ has independent elements, if $\ell\geq 2$, then all but finitely many $n\in S$ have distinct coordinates. Hence, Theorem 4.7 applies, and implies that in proving (26) we can assume that the system we work with is $(Z_{k},\mu_{k},T)$ for some $k\in{\mathbb{N}}$. Next, using Theorem 4.6 and the approximation property mentioned immediately after this result, we get that in proving Proposition 4.4 we can assume that the system $(X,\mu,T)$ is an ergodic $k$-step nilsystem. ∎

A very similar argument gives the following reduction.

Proposition 4.9**.**

If Proposition 4.5 holds for every ergodic nilsystem, then it holds for every ergodic system.

Proof.

First note that since the collection of sequences $a_{1},\ldots,a_{\ell}$ is weakly independent, if $\ell\geq 2$, then for all $n\in{\mathbb{N}}^{r}$ outside a set of density [math] the values $a_{1}(n),\ldots,a_{\ell}(n)$ are distinct. Hence, Theorem 4.7 applies and shows that in proving (27) we can assume that the system we work with is $(Z_{k},\mu_{k},T)$ for some $k\in{\mathbb{N}}$. We conclude the reduction as in the previous proposition. ∎

4.5. Proof of Propositions 4.4 and 4.5

Our plan is to first prove Propositions 4.4 and 4.5 in the case where the nilmanifold is Abelian and subsequently deal with the non-Abelian case.

Lemma 4.10**.**

Propositions  4.4 and 4.5 hold if $(X,\mu,T)$ is a rotation on a compact Abelian Lie group with the Haar measure.

Proof.

Suppose that $T$ is an ergodic rotation on a compact Abelian Lie group $X$. Then $X={\mathbb{Z}}_{u}\times{\mathbb{T}}^{v}$ for some $u\in{\mathbb{N}}$ and $v\in{\mathbb{Z}}_{+}$, where ${\mathbb{Z}}_{u}:={\mathbb{Z}}/(u{\mathbb{Z}})$. Moreover, we can assume that

[TABLE]

where addition is taken $\!\!\mod{u}$ on the first coordinate, and $\alpha=(\alpha_{1},\ldots,\alpha_{v})$ acts ergodically on ${\mathbb{T}}^{v}$, or equivalently, $1,\alpha_{1},\ldots,\alpha_{v}$ are rationally independent. Note also that $\mu=m_{X}=m_{{\mathbb{Z}}_{u}}\times m_{{\mathbb{T}}^{v}}$.

First, we prove Proposition 4.4. By approximation in $L^{2}(\mu)$ it suffices to verify (26) when

[TABLE]

where $x\in{\mathbb{Z}}_{u},y\in{\mathbb{T}}^{v},$ and $e(t):=e^{2\pi it}$ for $t\in{\mathbb{R}}$. Furthermore, since at least one of the functions $F_{0},\ldots,F_{\ell}$ is orthogonal to the rational Kronecker factor of the system, we can assume that $l_{j}\neq 0$ for some $j\in\{0,\ldots,\ell\}$. If $l_{1}=\cdots=l_{\ell}=0$, then $l_{0}\neq 0$, and in this case (recall that $n_{0}=0$)

[TABLE]

for every $n_{1},\ldots,n_{\ell}\in{\mathbb{N}}$, so (26) clearly holds. Suppose now that $l_{j}\neq 0$ for some $j\in\{1,\ldots,\ell\}$. Without loss of generality we can assume that $l_{1}\neq 0$.

Note that the limit in (26) is equal to

[TABLE]

For $n=(n_{1},\ldots,n_{\ell})\in{\mathbb{N}}^{\ell}$ let $\beta_{n}\in{\mathbb{T}}$ be defined by

[TABLE]

Since the set $S$ has independent elements, $\alpha=(\alpha_{1},\ldots,\alpha_{v})$ is such that $1,\alpha_{1},\ldots,\alpha_{v}$ are rationally independent, and $l_{1}\neq 0$, an easy computation shows that $\beta_{n}$ is irrational for all but finitely many $n\in S$. For those values of $n\in S$, it is well known that the sequence $(p\beta_{n})_{p\in{\mathbb{P}}_{d}}$ is equidistributed on ${\mathbb{T}}$. Hence, the bounded convergence theorem gives that the averages ${\mathbb{E}}_{{\mathbb{P}}_{d}}$ in (30) are [math] for all but finitely many $n\in S$, and as a consequence (26) holds.

Next, we prove Proposition 4.5. Again by approximation in $L^{2}(\mu)$, it suffices to verify (27) when

[TABLE]

where $x\in{\mathbb{Z}}_{u}$ and $y\in{\mathbb{T}}^{v}$. Suppose first that at least one of the $l_{0},\ldots,l_{\ell}$ is non-zero. As before, we can assume that $l_{1}\neq 0$. Then the right hand side in (27) is [math] and the left hand side is equal to

[TABLE]

As before, we argue that for all $n\in{\mathbb{N}}^{r}$ outside a set of density [math], the averages ${\mathbb{E}}_{p\in{\mathbb{P}}_{d}}$ are [math] and as a consequence (27) holds.

Suppose now that $l_{0}=\cdots=l_{\ell}=0$. If all $k_{0},\ldots,k_{\ell}$ are multiples of $u$, then (27) holds trivially. If not, as before, we can assume that $k_{1}$ is not a multiple of $u$. Then the right hand side in (27) is [math] and the left hand side is equal to

[TABLE]

If $\sum_{j=0}^{\ell}k_{j}$ is not a multiple of $u$, then all the integrals are [math]. Otherwise, the expression becomes

[TABLE]

The average over $p$ is finite (it is equal to the average over those $k\in\{0,\ldots,u-1\}$ that satisfy $(k,u)=1$ and $k\in d{\mathbb{N}}+1$), hence we can freely exchange the two averages and we get the expression

[TABLE]

If the last expression is non-zero, then there exists $p\in\{1,\ldots,u-1\}$ such that

[TABLE]

Since $k_{1}$ is not a multiple of $u$, this contradicts our assumption that the sequences $a_{1},\ldots,a_{\ell}$ are jointly equidistributed in congruence classes and completes the proof. ∎

The next result follows from Theorem 7 and Corollary 8 of Chapter 15 in [18] (see also [2, Theorem 5.4]) and was first established in a slightly different form in [33].

Lemma 4.11**.**

Let $\ell,n_{1},\ldots,n_{\ell}\in{\mathbb{N}}$. Let $X=G/\Gamma$ be a nilmanifold and $b\in G$ be an ergodic nilrotation. Then for $m_{X}$-almost every $x\in X$ the sequence

[TABLE]

*is equidistributed on a set $W_{x}=H\cdot\tilde{x}$, where $\tilde{x}:=(x,\ldots,x)$ and $H$ is a subgroup of $G^{\ell}$ such that $(g^{n_{1}},\ldots,g^{n_{\ell}})\in H$ for every $g\in G$. *

Remark**.**

We caution the reader that although for Abelian nilmanifolds the conclusion holds for every $x\in X$ this is not so for $s$-step nilmanifolds when $s\geq 2$.

The previous lemma is used in order to establish the following result:

Lemma 4.12**.**

Let $\ell\in{\mathbb{N}}$ and $S$ be a subset of ${\mathbb{N}}^{\ell}$ with independent elements. Let $s\in{\mathbb{N}}$, with $s\geq 2$, $X=G/\Gamma$ be an $s$-step nilmanifold, $b\in G$ be an ergodic nilrotation, and $\Phi_{1},\ldots,\Phi_{\ell}$ be $s$-step nilcharacters of $X$, at least one of which is non-trivial. Then for all but finitely many $(n_{1},\ldots,n_{\ell})\in S$ we have

[TABLE]

for $m_{X}$-almost every $x\in X$.

Proof.

Henceforth, with $n_{1},\ldots,n_{\ell}$ we denote the coordinates of $n\in{\mathbb{N}}^{\ell}$. By Lemma 4.11 there exists $X^{\prime}\subset X$ with $m_{X}(X^{\prime})=1$ such that for every $x\in X^{\prime}$ and every $n\in{\mathbb{N}}^{\ell}$ we have

[TABLE]

where $W_{x,n}=H_{n}\,\tilde{x}$, $\tilde{x}:=(x,\ldots,x)$, and $H_{n}$ is a subgroup of $G^{\ell}$ such that $(g^{n_{1}},\ldots,g^{n_{\ell}})\in H_{n}$ for every $g\in G$. In particular, for every $x\in X^{\prime}$, we have that

[TABLE]

Since $s\geq 2$, as remarked in Section 4.3 we can assume that $G_{s}={\mathbb{T}}^{t}$ for some $t\in{\mathbb{N}}$ ($G_{s}$ is non-trivial since $X$ supports a non-trivial $s$-step nilcharacter). Henceforth, we use additive notation for elements of $G_{s}$. By assumption, for $j=1,\ldots,\ell$ we have

[TABLE]

where $k_{j}\in{\mathbb{Z}}^{t}$. Since at least one of the nilcharacters is non-trivial, we can assume that $\Phi_{1}$ is, or equivalently, that $k_{1}\in{\mathbb{Z}}^{t}$ is non-zero. Using this, the translation invariance of the measure $m_{W_{x,n}}$, and (33), (34), we deduce that for every $x\in X^{\prime}$ and $n\in{\mathbb{N}}^{\ell}$ we have

[TABLE]

unless $\sum_{j=1}^{\ell}n_{j}\,(k_{j}\cdot u)=0\pmod{1}$ for every $u\in{\mathbb{T}}^{t}$. Equivalently, writing $k_{j}=(k_{j,1},\ldots,k_{j,t})$, $j=1,\ldots,\ell$, and $u=(u_{1},\ldots,u_{t})$, we get that if the previous sum is zero, then

[TABLE]

Hence,

[TABLE]

Since the integers $k_{1,1},\ldots,k_{1,\ell}$ are not all zero (recall that $k_{1}\neq 0$), and the set $S$ has independent elements, we get that the first identity of the previous system can be satisfied only for finitely many $n\in S$. Hence, for all but finitely many $n\in S$, for every $x\in X^{\prime}$ equation (35) holds, and as a consequence (32) holds. This completes the proof. ∎

We will also use the following result, which is proved using the Gowers uniformity of the modified von Mangoldt function [14]. It is proved in [7, Theorem 4.4] for $d=1$ but the same argument gives the proof for general $d\in{\mathbb{N}}$.

Proposition 4.13**.**

Let $d,r_{0}\in{\mathbb{N}}$ and $(X,\mu,T)$ be a system such that the ergodic components of the system $(X,\mu,T^{r_{0}})$ are totally ergodic. Let

[TABLE]

Then all the limits below exist and we have

[TABLE]

for all $\ell\in{\mathbb{N}}$ and $F_{0},\ldots,F_{\ell}\in L^{\infty}(\mu)$.

Proof of Propositions  4.4 and 4.5.

By Propositions 4.8 and 4.9 we can assume that the system is an ergodic nilsystem. So let $X=G/\Gamma$ be an $s$-step nilmanifold with the Haar measure $m_{X}$ and $Tx=bx$, $x\in X$, for some $b\in G$.

We prove the statements by induction on $s\in{\mathbb{N}}$ . If $s=1$, then $X$ is a compact Abelian Lie group so we are covered by Lemma 4.10.

Suppose that $s\geq 2$ and Propositions  4.4 and 4.5 hold for all $(s-1)$-step nilsystems. Since linear combinations of $s$-step nilcharacters are dense in $C(X)$, using an approximation argument, we can assume that for $j=0,\ldots,\ell$ we have $F_{j}=\Phi_{j}$ where $\Phi_{j}$ is an $s$-step nilcharacter of $X$. If $\Phi_{0},\ldots,\Phi_{\ell}$ are all trivial $s$-step nilcharacters of $X$, then they factorize through the nilmanifold

[TABLE]

The group $G/G_{s}$ is $(s-1)$-step nilpotent and $X^{\prime}$ is an $(s-1)$-step nilmanifold. So in this case the result follows from the induction hypothesis. Hence, we can assume that at least one of the $s$-step nilcharacters $\Phi_{0},\ldots,\Phi_{\ell}$ is non-trivial.

Suppose first that $\Phi_{0}$ is a non-trivial $s$-step nilcharacter and $\Phi_{1},\ldots,\Phi_{\ell}$ are trivial $s$-step nilcharacters. Then for every $n_{0},\ldots,n_{\ell}\in{\mathbb{Z}}$ the function $\prod_{j=0}^{\ell}T^{pn_{j}}\Phi_{j}$ is also a non-trivial $s$-step nilcharacter (with the same frequency as $\Phi_{0}$). Hence, $\int\Phi_{0}(x)\,dm_{X}=0$ and

[TABLE]

In this case, equations (26) and (27) clearly hold.

Hence, without loss of generality, we can assume that $\Phi_{1}$ is a non-trivial $s$-step nilcharacter. In this case we prove Proposition 4.4, the proof of Proposition 4.5 is very similar. We have to show that

[TABLE]

Since $(X,\mu,T)$ is an ergodic nilsystem, there exists $r_{0}\in{\mathbb{N}}$ such that $T^{r_{0}}$ acts ergodically on $X^{0}$ and the ergodic components of $(X,\mu,T^{r_{0}})$ are totally ergodic (see for example Corollaries 7 and 8 on page 182 in [18]). By Proposition 4.13 it suffices to show that for every $k\in{\mathbb{N}}$ we have

[TABLE]

Recall that $n_{0}=0$. By the bounded convergence theorem (from [20] the limits ${\mathbb{E}}_{m\in{\mathbb{N}}}$ exist for every $x\in X$) it suffices to show that for every $k\in{\mathbb{N}}$, for $m_{X}$-almost every $x\in X$, we have

[TABLE]

Hence, it suffices to show that for every $k\in{\mathbb{N}}$, for $m_{X}$-almost every $x\in X$, we have

[TABLE]

where $b^{\prime}:=b^{dr_{0}}$ and for $j=1,\ldots,\ell$ we let $\Phi_{j}^{\prime}(x):=\Phi_{j}(b^{kn_{j}}x)$ for $x\in X$. Then $b^{\prime}$ acts ergodically on $X_{0}$ and as explained in Section 4.3, for $j=1,\ldots,\ell$, the restriction of $\Phi^{\prime}_{j}$ to $X_{0}$ is an $s$-step nilcharacter of $X_{0}$ with same frequency as $\Phi_{j}$; hence, $\Phi_{1}$ is non-trivial. The validity of the last identity follows from Lemma 4.12 (here we used crucially that $s\geq 2$). This completes the proof. ∎