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Originating in harmonic analysis, interpolation sets were first studied in dynamics by Glasner and Weiss in the 1980s. A set $S \subset \mathbb{N}$ is an interpolation set for a class of topological dynamical systems $\mathcal{C}$ if any bounded sequence on $S$ can be extended to a sequence that arises from a system in $\mathcal{C}$. In this paper, we provide combinatorial characterizations of interpolation sets for: $\bullet$ (totally) minimal systems; $\bullet$ topologically (weak) mixing systems; $\bullet$ strictly ergodic systems; and $\bullet$ zero entropy systems. Additionally, we prove some results on a slightly different notion, called weak interpolation sets, for several classes of systems. We also answer a question of Host, Kra, and Maass concerning the connection between sets of pointwise recurrence for distal systems and $IP$-sets., Comment: 31 pages, to appear in Trans. Amer. Math. Soc

In contrast to finite arithmetic configurations, relatively little is known about which infinite patterns can be found in every set of natural numbers with positive density. Building on recent advances showing infinite sumsets can be found, we explore numerous open problems and obstructions to finding other infinite configurations in every set of natural numbers with positive density., Comment: 37 pages

Let $(G/\Gamma,R_a)$ be an ergodic $k$-step nilsystem for $k\geq 2$. We adapt an argument of Parry to show that $L^2(G/\Gamma)$ decomposes as a sum of a subspace with discrete spectrum and a subspace of Lebesgue spectrum with infinite multiplicity. In particular, we generalize a result previously established by Host, Kra and Maass for $2$-step nilsystems and a result by Stepin for nilsystems $G/\Gamma$ with connected, simply connected $G$., Comment: 12 pages

We show that every set $A$ of natural numbers with positive upper density can be shifted to contain the restricted sumset $\{b_1 + b_2 : b_1, b_2\in B \text{ and } b_1 \neq b_2 \}$ for some infinite set $B \subset A$., Comment: 14 pages. Changes after referee comments, improved exposition

Motivated by questions asked by Erdos, we prove that any set $A\subset{\mathbb N}$ with positive upper density contains, for any $k\in{\mathbb N}$, a sumset $B_1+\cdots+B_k$, where $B_1,\dots,B_k\subset{\mathbb N}$ are infinite. Our proof uses ergodic theory and relies on structural results for measure preserving systems. Our techniques are new, even for the previously known case of $k=2$., Comment: 49 Pages

We give a new proof of a sumset conjecture of Furstenberg that was first proved by Hochman and Shmerkin in 2012: if $\log r / \log s$ is irrational and $X$ and $Y$ are $\times r$- and $\times s$-invariant subsets of $[0,1]$, respectively, then $\dim_\text{H} (X+Y) = \min ( 1, \dim_\text{H} X + \dim_\text{H} Y)$. Our main result yields information on the size of the sumset $\lambda X + \eta Y$ uniformly across a compact set of parameters at fixed scales. The proof is combinatorial and avoids the machinery of local entropy averages and CP-processes, relying instead on a quantitative, discrete Marstrand projection theorem and a subtree regularity theorem that may be of independent interest., Comment: 26 pages. Some of this work was originally posted to the arXiv in the first half of the paper "Additive transversality of fractal sets in the reals and the integers" (arXiv:2007.05480v1); that paper has since been updated (arXiv:2007.05480) and no longer includes any of this work

Katznelson's Question is a long-standing open question concerning recurrence in topological dynamics with strong historical and mathematical ties to open problems in combinatorics and harmonic analysis. In this article, we give a positive answer to Katznelson's Question for certain towers of skew product extensions of equicontinuous systems, including systems of the form $(x,t) \mapsto (x + \alpha, t + h(x))$. We describe which frequencies must be controlled for in order to ensure recurrence in such systems, and we derive combinatorial corollaries concerning the difference sets of syndetic subsets of the natural numbers., Comment: 38 pages, revised version; the statement of Theorem C has become more general addressing the case in which a higher-order derivative of the sequence f is almost periodic. To appear in BAMS

By juxtaposing ideas from fractal geometry and dynamical systems, Furstenberg proposed a series of conjectures in the late 1960's that explore the relationship between digit expansions with respect to multiplicatively independent bases. In this work, we introduce and study - in the discrete context of the integers - analogues of some of the notions and results surrounding Furstenberg's work. In particular, we define a new class of fractal sets of integers that parallels the notion of $\times r$-invariant sets on the 1-torus and investigate the additive and geometric independence between two such fractal sets when they are structured with respect to multiplicatively independent bases. Our main results in this direction parallel the works of Furstenberg, Hochman-Shmerkin, Shmerkin, Wu, and Lindenstrauss-Meiri-Peres and include: -a classification of all subsets of the positive integers that are simultaneously $\times r$- and $\times s$-invariant; -integer analogues of two of Furstenberg's transversality conjectures pertaining to the dimensions of the intersection $A\cap B$ and the sumset $A+B$ of $\times r$- and $\times s$-invariant sets $A$ and $B$ when $r$ and $s$ are multiplicatively independent; and -a description of the dimension of iterated sumsets $A+A+\cdots+A$ for any $\times r$-invariant set $A$. We achieve these results by combining ideas from fractal geometry and ergodic theory to build a bridge between the continuous and discrete regimes. For the transversality results, we rely heavily on quantitative bounds on the $L^q$-dimensions of projections of restricted digit Cantor measures obtained recently by Shmerkin. We end by outlining a number of open questions and directions regarding fractal subsets of the integers., Comment: 51 pages

We obtain new results pertaining to convergence and recurrence of multiple ergodic averages along functions from a Hardy field. Among other things, we confirm some of the conjectures posed by Frantzikinakis in [Fra10; Fra16] and obtain combinatorial applications which contain, as rather special cases, several previously known (polynomial and non-polynomial) extensions of Szemeredi's theorem on arithmetic progressions [BL96; BLL08; FW09; Fra10; BMR17]. One of the novel features of our results, which is not present in previous work, is that they allow for a mixture of polynomials and non-polynomial functions. As an illustration, assume $f_i(t)=a_{i,1}t^{c_{i,1}}+\cdots+a_{i,d}t^{c_{i,d}}$ for $c_{i,j}>0$ and $a_{i,j}\in\mathbb{R}$. Then $\bullet$ for any measure preserving system $(X,{\mathcal B},\mu,T)$ and $h_1,\dots,h_k\in L^\infty(X)$, the limit $$\lim_{N\to\infty}\frac{1}{N}\sum_{n=1}^N T^{[f_1(n)]}h_1\cdots T^{[f_k(n)]}h_k$$ exists in $L^2$; $\bullet$ for any $E\subset \mathbb{N}$ with $\overline{\mathrm{d}}(E)>0$ there are $a,n\in\mathbb{N}$ such that $\{a,\, a+[f_1(n)],\ldots,a+[f_k(n)]\}\subset E$. We also show that if $f_1,\dots,f_k$ belong to a Hardy field, have polynomial growth, and are such that no linear combination of them is a polynomial, then for any measure preserving system $(X,{\mathcal B},\mu,T)$ and any $A\in{\mathcal B}$, $$\limsup_{N\to\infty}\frac{1}{N}\sum_{n=1}^N\mu\Big(A\cap T^{-[ f_1(n) ]}A\cap\ldots\cap T^{-[f_k(n)]}A\Big)\,\geq\,\mu(A)^{k+1}.$$, Comment: 41 pages, 2nd revised version implementing referee comments

We study equidistribution properties of translations on nilmanifolds along functions of polynomial growth from a Hardy field. More precisely, if $X=G/\Gamma$ is a nilmanifold, $a_1,\ldots,a_k\in G$ are commuting nilrotations, and $f_1,\ldots,f_k$ are functions of polynomial growth from a Hardy field then we show that $\bullet$ the distribution of the sequence $a_1^{f_1(n)}\cdot\ldots\cdot a_k^{f_k(n)}\Gamma$ is governed by its projection onto the maximal factor torus, which extends Leibman's Equidistribution Criterion form polynomials to a much wider range of functions; and $\bullet$ the orbit closure of $a_1^{f_1(n)}\cdot\ldots\cdot a_k^{f_k(n)}\Gamma$ is always a finite union of sub-nilmanifolds, which extends some of the previous work of Leibman and Frantzikinakis on this topic., Comment: 49 pages, 2nd revised version after referee comments

Let $T$ be a measure preserving $\mathbb{Z}^\ell$-action on the probability space $(X,{\mathcal B},\mu),$ $q_1,\dots,q_m:{\mathbb R}\to{\mathbb R}^\ell$ vector polynomials, and $f_0,\dots,f_m\in L^\infty(X)$. For any $\epsilon > 0$ and multicorrelation sequences of the form $\displaystyle\alpha(n)=\int_Xf_0\cdot T^{ \lfloor q_1(n) \rfloor }f_1\cdots T^{ \lfloor q_m(n) \rfloor }f_m\;d\mu$ we show that there exists a nilsequence $\psi$ for which $\displaystyle\lim_{N - M \to \infty} \frac{1}{N-M} \sum_{n=M}^{N-1} |\alpha(n) - \psi(n)| \leq \epsilon$ and $\displaystyle\lim_{N \to \infty} \frac{1}{\pi(N)} \sum_{p \in {\mathbb P}\cap[1,N]} |\alpha(p) - \psi(p)| \leq \epsilon.$ This result simultaneously generalizes previous results of Frantzikinakis [2] and the authors [11,13]., Comment: 7 pages

We establish two ergodic theorems which have among their corollaries numerous classical results from multiplicative number theory, including the Prime Number Theorem, a theorem of Pillai-Selberg, a theorem of Erd\H{o}s-Delange, the mean value theorem of Wirsing, and special cases of the mean value theorem of Hal\'asz. By building on the ideas behind our ergodic results, we recast Sarnak's M\"obius disjointness conjecture in a new dynamical framework. This naturally leads to an extension of Sarnak's conjecture which focuses on the disjointness of additive and multiplicative semigroup actions. We substantiate this extension by providing proofs of several special cases., Comment: 56 pages, implemented changes following the referees comments

Let $\Omega(n)$ denote the number of prime factors of $n$. We show that for any bounded $f\colon\mathbb{N}\to\mathbb{C}$ one has \[ \frac{1}{N}\sum_{n=1}^N\, f(\Omega(n)+1)=\frac{1}{N}\sum_{n=1}^N\, f(\Omega(n))+\mathrm{o}_{N\to\infty}(1). \] This yields a new elementary proof of the Prime Number Theorem., Comment: 13 pages; to appear in Bulletin of the London Mathematical Society; corrected a small mistake in the proof of Proposition 2.2, see footnote 2 at the bottom of page 10

We study multicorrelation sequences arising from systems with commuting transformations. Our main result is a refinement of a decomposition result of Frantzikinakis and it states that any multicorrelation sequences for commuting transformations can be decomposed, for every $\epsilon>0$, as the sum of a nilsequence $\phi(n)$ and a sequence $\omega(n)$ satisfying $\lim_{N\to\infty}\frac{1}{N}\sum_{n=1}^N |\omega(n)|<\epsilon$ and $\lim_{N\to\infty}\frac{1}{|\mathbb{P}\cap [N]|}\sum_{p\in \mathbb{P}\cap [N]} |\omega(p)|<\epsilon$., Comment: 27 pages

In this work we study the set of eventually always hitting points in shrinking target systems. These are points whose long orbit segments eventually hit the corresponding shrinking targets for all future times. We focus our attention on systems where translates of targets exhibit near perfect mutual independence, such as Bernoulli schemes and the Gauss map. For such systems, we present tight conditions on the shrinking rate of the targets so that the set of eventually always hitting points is a null set (or co-null set respectively)., Comment: 29 pages, revised version with fixed typos, added remarks, and other small changes

We investigate the relationship between the dynamical properties of minimal topological dynamical systems and the multiplicative combinatorial properties of return time sets arising from those systems. In particular, we prove that for a residual sets of points in any minimal system, the set of return times to any non-empty, open set contains arbitrarily long geometric progressions. Under the separate assumptions of total minimality and distality, we prove that return time sets have positive multiplicative upper Banach density along $\mathbb{N}$ and along multiplicative subsemigroups of $\mathbb{N}$, respectively. The primary motivation for this work is the long-standing open question of whether or not syndetic subsets of the positive integers contain arbitrarily long geometric progressions; our main result is some evidence for an affirmative answer to this question., Comment: 32 pages

We establish new recurrence and multiple recurrence results for a rather large family $\mathcal{F}$ of non-polynomial functions which includes tempered functions defined in [11], as well as functions from a Hardy field with the property that for some $\ell\in \mathbb{N}\cup\{0\}$, $\lim_{x\to\infty }f^{(\ell)}(x)=\pm\infty$ and $\lim_{x\to\infty }f^{(\ell+1)}(x)=0$. Among other things, we show that for any $f\in\mathcal{F}$, any invertible probability measure preserving system $(X,\mathcal{B},\mu,T)$, any $A\in\mathcal{B}$ with $\mu(A)>0$, and any $\epsilon>0$, the sets of returns $$ R_{\epsilon, A}= \big\{n\in\mathbb{N}:\mu(A\cap T^{-\lfloor f(n)\rfloor}A)>\mu^2(A)-\epsilon\big\} $$ and $$ R^{(k)}_{A}= \big\{ n\in\mathbb{N}: \mu\big(A\cap T^{\lfloor f(n)\rfloor}A\cap T^{\lfloor f(n+1)\rfloor}A\cap\cdots\cap T^{\lfloor f(n+k)\rfloor}A\big)>0\big\} $$ possess somewhat unexpected properties of largeness; in particular, they are thick, i.e., contain arbitrarily long intervals., Comment: 51 pages

Given a level set $E$ of an arbitrary multiplicative function $f$, we establish, by building on the fundamental work of Frantzikinakis and Host [13,14], a structure theorem which gives a decomposition of $\mathbb{1}_E$ into an almost periodic and a pseudo-random parts. Using this structure theorem together with the technique developed by the authors in [3], we obtain the following result pertaining to polynomial multiple recurrence. Let $E=\{n_10$, all $\ell\geq 1$ and all polynomials $p_i\in\mathbb{Z}[x]$, $i=1,\ldots,\ell$, with $p_i(0)=0$ we have $$ \lim_{N\to\infty}\frac{1}{N}\sum_{j=1}^N \mu\big(A\cap T^{-p_1(n_j)}A\cap\ldots\cap T^{-p_\ell(n_j)}A\big)>0. $$ We also show that if a level set $E$ of a multiplicative function has positive upper density, then any self-shift $E-r$, $r\in E$, is a set of averaging polynomial multiple recurrence. This in turn leads to the following refinement of the polynomial Szemer\'edi theorem (cf. [4]). Let $E$ be a level set of an arbitrary multiplicative function, suppose $E$ has positive upper density and let $r\in E$. Then for any set $D\subset \mathbb{N}$ with positive upper density and any polynomials $p_i\in\mathbb{Q}[t]$, $i=1,\ldots,\ell$, which satisfy $p_i(\mathbb{Z})\subset\mathbb{Z}$ and $p_i(0)=0$ for all $i\in\{1,\ldots,\ell\}$, there exists $\beta>0$ such that the set $$ \left\{\,n\in E-r:\overline{d}\Big(D\cap (D-p_1(n))\cap \ldots\cap(D-p_\ell(n)) \Big)>\beta \,\right\} $$ has positive lower density., Comment: 32 pages. Formerly part of arXiv:1705.07322

We generalize Berg's notion of quasi-disjointness to actions of countable groups and prove that every measurably distal system is quasi-disjoint from every measure preserving system. As a corollary we obtain easy to check necessary and sufficient conditions for two systems to be disjoint, provided one of them is measurably distal. We also obtain a Wiener--Wintner type theorem for countable amenable groups with distal weights and applications to weighted multiple ergodic averages and multiple recurrence., Comment: 28 pages. v2: Small modifications in Section 6 as it previously used an incorrect result from [MR19] that has since been corrected in [MR21] (arxiv.org/abs/1609.03631)

A set $R\subset \mathbb{N}$ is called rational if it is well-approximable by finite unions of arithmetic progressions. Examples of rational sets include many classical sets of number-theoretical origin such as the set of squarefree numbers, the set of abundant numbers, or sets of the form $\Phi_x:=\{n\in\mathbb{N}: \frac{\boldsymbol{\varphi}(n)}{n}0$, then the following are equivalent: (a) $R$ is divisible, i.e. $\overline{d}(R\cap u \mathbb{N})>0$ for all $u\in\mathbb{N}$. (b) $R$ is an averaging set of polynomial single recurrence. (c) $R$ is an averaging set of polynomial multiple recurrence. As an application, we show that if $R$ is rational and divisible, then for any set $E\subset\mathbb{N}$ with $\overline{d}(E)>0$ and any polynomials $p_i\in\mathbb{Q}[t]$,$i=1,\ldots,\ell$, which satisfy $p_i(\mathbb{Z})\subset\mathbb{Z}$ and $p_i(0)=0$ for all $i\in\{1,\ldots,\ell\}$, there exists $\beta>0$ such that the set $$\{n\in R:\overline{d}( E\cap (E-p_1(n))\cap\ldots\cap(E-p_\ell(n)))>\beta\}$$ has positive lower density. Ramsey-theoretical applications naturally lead to problems in symbolic dynamics, which involve rationally almost periodic sequences. We prove that if $\mathcal{A}$ is a finite alphabet, $\eta\in\mathcal{A}^\mathbb{N}$ is rationally almost periodic, $S$ denotes the left-shift on $\mathcal{A}^\mathbb{Z}$ and $$X:=\{y\in \mathcal{A}^\mathbb{Z} : \text{each finite word appearing in $y$ appears in }\eta\},$$ then $\eta$ is a generic point for an $S$-invariant probability measure $\nu$ on $X$ such that $(X,\nu,S)$ is ergodic and has rational discrete spectrum., Comment: 53 pages

We investigate how spectral properties of a measure preserving system $(X,\mathcal{B},\mu,T)$ are reflected in the multiple ergodic averages arising from that system. For certain sequences $a:\mathbb{N}\to\mathbb{N}$ we provide natural conditions on the spectrum $\sigma(T)$ such that for all $f_1,\ldots,f_k\in L^\infty$, \begin{equation*} \lim_{N\to\infty} \frac{1}{N} \sum_{n=1}^N \prod_{j=1}^k T^{ja(n)}f_j = \lim_{N\rightarrow\infty} \frac{1}{N} \sum_{n=1}^N \prod_{j=1}^k T^{jn}f_j \end{equation*} in $L^2$-norm. In particular, our results apply to infinite arithmetic progressions $a(n)=qn+r$, Beatty sequences $a(n)=\lfloor \theta n+\gamma\rfloor$, the sequence of squarefree numbers $a(n)=q_n$, and the sequence of prime numbers $a(n)=p_n$. We also obtain a new refinement of Szemer\'edi's theorem via Furstenberg's correspondence principle. ERRATUM: Theorem 7.1 in the paper is incorrect as stated, and the error originates with Proposition 7.5, part (iii), which was incorrectly quoted from the literature. In the attached erratum we fix the problem by establishing a slightly weaker version of Theorem 7.1 and use it to give a new proof of Theorem 4.2. This ensures that all main results in our main article remain correct. We thank Zhengxing Lian and Jiahao Qiu for bringing this mistake to our attention., Comment: 32 page article + 5 page erratum