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We develop a robust structure theory for multiple ergodic averages of commuting transformations along Hardy sequences of polynomial growth. We then apply it to derive a number of novel results on joint ergodicity, recurrence and convergence. In particular, we prove joint ergodicity for (a) pairwise independent Hardy sequences and weakly mixing transformations, (b) strongly independent Hardy sequences and ergodic transformations, (c) strongly irrationally independent Hardy sequences and totally ergodic transformations. We use these joint ergodicity results to provide new recurrence results for multidimensional patterns along strongly independent Hardy sequences, showing for instance that all subsets of $\mathbb{Z}^2$ of positive upper density contain patterns \[ (m_1, m_2),\; (m_1 + \lfloor{n^{3/2}}\rfloor, m_2),\; (m_1, m_2 + \lfloor{n^{3/2} + n^{1/2}}\rfloor).\] Last but not least, we positively resolve the joint ergodicity classification problem for pairwise independent Hardy sequences, of which the aforementioned families are special cases. While building on recent technical advances (e.g. PET coefficient tracking schemes and joint ergodicity criteria), our work introduces a number of technical developments of its own. We construct a suitable generalization of Host-Kra and box seminorms that quantitatively control ergodic averages along Hardy sequences. We subsequently use them to obtain Host-Kra seminorm estimates for averages along all pairwise independent Hardy sequences. Furthermore, we develop an ergodic version of the quantitative concatenation argument that has recently found extensive use in combinatorics, number theory and harmonic analysis. Lastly, we improve simultaneous Taylor approximations for Hardy sequences., Comment: 105 pages. Comments welcome!

We establish sufficient and necessary conditions for the joint transitivity of linear iterates in a minimal topological dynamical system with commuting transformations. This result provides the first topological analogue of the classical Berend and Bergelson joint ergodicity criterion in measure-preserving systems., Comment: Comments welcome!

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

We investigate the limiting behavior of multiple ergodic averages along sparse sequences evaluated at prime numbers. Our sequences arise from smooth and well-behaved functions that have polynomial growth. Central to this topic is a comparison result between standard Ces\'{a}ro averages along positive integers and averages weighted by the (modified) von Mangoldt function. The main ingredients are a recent result of Matom\"{a}ki, Shao, Tao and Ter\"{a}v\"{a}inen on the Gowers uniformity of the latter function in short intervals, a lifting argument that allows one to pass from actions of integers to flows, a simultaneous (variable) polynomial approximation in appropriate short intervals, and some quantitative equidistribution results for the former polynomials. We derive numerous applications in multiple recurrence, additive combinatorics, and equidistribution in nilmanifolds along primes. In particular, we deduce that any set of positive density contains arithmetic progressions with step $\lfloor p^c \rfloor$, where $c$ is a positive non-integer and $p$ denotes a prime, establishing a conjecture of Frantzikinakis., Comment: 58 pages

The metal-insulator transition (MIT) observed in vanadium dioxide (VO2) has been a topic of great research interest for past decades, with the underlying physics yet not fully understood due to the complex electron interactions and structures involved. The ability to understand and tune the MIT behaviour is of vital importance from the perspective of both underlying fundamental science as well as potential applications. In this work, we use scanning transmission electron microscopy (STEM) to investigate cross-section lamella of the VO2 films deposited using pulsed laser deposition (PLD) on three substrates: c-cut sapphire, TiO2(101) and TiO2(001). Advanced STEM imaging is performed in which also the oxygen atom columns are resolved. We link the overall film quality and structures on atomic and nanoscale to the electrical transition characteristics. We observe poor MIT characteristics on c-sapphire due to the presence of very small domains with six orientation variants, and on TiO2 (001) due to the presence of cracks induced by stress relaxation. However, the MIT on TiO2 (101) behaves favourably, despite similar stress relaxation which, however, only lead to domain boundaries but no cracks.

Examining multiple ergodic averages whose iterates are integer parts of real valued polynomials for totally ergodic systems, we provide various characterizations of total joint ergodicity, meaning that an average converges to the "expected" limit along every arithmetic progression. In particular, we obtain a complete characterization when the number of iterates is at most two, and disprove a conjecture of the first author. We also improve a result of Frantzikinakis on joint ergodicity of Hardy field functions of at most polynomial growth for totally ergodic systems, which extends a conjecture of Bergelson-Moreira-Richter. Our method is to first use the methodology of Frantzikinakis, which allows one to reduce the systems to rotations on abelian groups without using deep tools from ergodic theory, then develop formulas for integrals of exponential functions over subtori, and finally, compute exponential sums for integer parts of real polynomials., Comment: 30 pages

We provide necessary and sufficient conditions for joint ergodicity results for systems of commuting measure preserving transformations for an iterated Hardy field function of polynomial growth. Our method builds on and improves recent techniques due to Frantzikinakis and Tsinas, who dealt with multiple ergodic averages along Hardy field functions; it also enhances an approach introduced by the authors and Ferr\'e Moragues to study polynomial iterates. The more general expression, in which the iterate is a linear combination of a Hardy field function of polynomial growth and a tempered function, is studied as well., Comment: 36 pages; Theorem 6.1 now covers the case where the iterate is a linear combination of a Hardy field and a tempered function of different polynomial growth rates, extending in particular Theorem 1.1

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

We show that, under finitely many ergodicity assumptions, any multicorrelation sequence defined by invertible measure preserving $\mathbb{Z}^d$-actions with multivariable integer polynomial iterates is the sum of a nilsequence and a null sequence, extending a recent result of the second author. To this end, we develop a new seminorm bound estimate for multiple averages by improving the results in a previous work of the first, third and fourth authors. We also use this approach to obtain new criteria for joint ergodicity of multiple averages with multivariable polynomial iterates on $\mathbb{Z}^{d}$-systems., Comment: comments welcome!

In this paper we study multiple ergodic averages for "good" variable polynomials. In particular, under an additional assumption, we show that these averages converge to the expected limit, making progress related to an open problem posted by Frantzikinakis (Problem 10, "Some open problems on multiple ergodic averages. Bulletin of the Hellenic Mathematical Society. 60 (2016), 41-90"). Corresponding averages along prime numbers are studied too. These general convergence results imply various variable extensions of classical recurrence, combinatorial and number theoretical results which are presented as well., Comment: 32 pages. To appear in Discrete and Continuous Dynamical Systems. For the corresponding results along prime numbers, check v1

Following Frantzikinakis' approach on averages for Hardy field functions of different growth, we add to the topic by studying the corresponding averages for tempered functions, a class which also contains functions that oscillate and is in general more restrictive to deal with. Our main result is the existence and the explicit expression of the $L^2$-norm limit of the aforementioned averages, which turns out, as in the Hardy field case, to be the "expected" one. The main ingredients are the use of, the now classical, PET induction (introduced by Bergelson), covering a more general case, namely a "nice" class of tempered functions (developed by Chu-Frantzikinakis-Host for polynomials and Frantzikinakis for Hardy field functions) and some equidistribution results on nilmanifolds (analogous to the ones of Frantzikinakis' for the Hardy field case)., Comment: 31 pages. To appear in Discrete and Continuous Dynamical Systems

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

Exploiting the recent work of Tao and Ziegler on a concatenation theorem on factors, we find explicit characteristic factors for multiple averages along polynomials on systems with commuting transformations, and use them to study criteria of joint ergodicity for sequences of the form $(T^{p_{1,j}(n)}_{1}\cdot\ldots\cdot T^{p_{d,j}(n)}_{d})_{n\in\mathbb{Z}},$ $1\leq j\leq k$, where $T_{1},\dots,T_{d}$ are commuting measure preserving transformations on a probability measure space and $p_{i,j}$ are integer polynomials. To be more precise, we provide a sufficient condition for such sequences to be jointly ergodic, giving also a characterization for sequences of the form $(T^{p(n)}_{i})_{n\in\mathbb{Z}}, 1\leq i\leq d$ to be jointly ergodic, answering a question due to Bergelson., Comment: Some changes have been made correcting minor issues

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

For any measure preserving system $(X,\mathcal{B},\mu,T_1,\ldots,T_d),$ where we assume no commutativity on the transformations $T_i,$ $1\leq i\leq d,$ we study the pointwise convergence of multiple ergodic averages with iterates of different growth coming from a large class of sublinear functions. This class properly contains important subclasses of Hardy field functions of order $0$ and of Fej\'er functions, i.e., tempered functions of order $0.$ We show that the convergence of the single average, via an invariant property, implies the convergence of the multiple one. We also provide examples of sublinear functions which are in general bad for convergence on arbitrary systems, but they are good for uniquely ergodic systems. The case where the fastest function is linear is addressed as well, and we provide, in all the cases, an explicit formula of the limit function.

Exploiting the equidistribution properties of polynomial sequences, following the methods developed by Leibman ("Pointwise Convergence of ergodic averages for polynomial sequences of translations on a nilmanifold. Ergodic Theory Dynam. Systems, 25 (2005) no. 1, 201-213") and Frantzikinakis ("Multiple recurrence and convergence for Hardy field sequences of polynomial growth. Journal d'Analyse Mathematique, 112 (2010), 79-135" and "Equidistribution of sparse sequences on nilmanifolds. Journal d'Analyse Mathematique, 109 (2009), 353-395") we show that the ergodic averages with iterates given by the integer part of real-valued strongly independent polynomials, converge in the mean to the "right"-expected limit. These results have, via Furstenberg's correspondence principle, immediate combinatorial applications while combining these results with methods from "The polynomial multidimensional Szemer\'edi theorem along shifted primes. Israel J. Math., 194 (2013), no. 1, 331-348" and "Closest integer polynomial multiple recurrence along shifted primes. Ergodic Theory Dynam. Systems, 1-20. doi:10.1017/etds.2016.40" we get the respective "right" limits and combinatorial results for multiple averages for a single sequence as well as for several sequences along prime numbers., Comment: 18 pages

Following an approach presented by N. Frantzikinakis, B. Host and B. Kra, we show that the parameters in the multidimensional Szemer\'edi theorem for closest integer polynomials have non-empty intersection with the set of shifted primes $\mathbb{P}-1$ (or similarly of $\mathbb{P}+1$). Using the Furstenberg Correspondence Principle, we show this result by recasting it as a polynomial multiple recurrence result in measure ergodic theory. Furthermore, we obtain integer part polynomial convergence results by the same method, which is a transference principle that enables one to deduce results for $\mathbb{Z}$-actions from results for flows. We also give some applications of our approach on Gowers uniform sets., Comment: 18 pages; Changes in Sections 1,3,4 and 5

Following an approach presented by N. Frantzikinakis, we prove that any multiple correlation sequence, defined by invertible measure preserving actions of commuting transformations with integer part polynomial iterates, is the sum of a nilsequence and an error term, small in uniform density. As an intermediate result, we show that multiple ergodic averages with iterates given by the integer part of real valued polynomials converge in the mean. Also, we show that under certain assumptions the limit is zero. An important role in our arguments plays a transference principle, communicated to us by M. Wierdl, that enables to deduce results for $\mathbb{Z}$-actions from results for flows., Comment: 23 pages. Changes from v1: Added references and a new section, addressing some special cases where the limit of multiple ergodic averages with iterates given by the integer part of real valued polynomials is zero. Minor changes because of this in some previous sections as well. Changes from v2: Minor changes in the formulation of Theorem 5.1

Introducing the notion of a rational system of measure preserving transformations and proving a recurrence result for such systems, we give sufficient conditions in order a subset of rational numbers to contain arbitrary long arithmetic progressions., Comment: 8 pages. arXiv admin note: text overlap with arXiv:1212.3520

We prove additive and multiplicative partition theorems, obtaining combinatorial results for p-quasicyclic groups, where p is a prime number. We also get density results for p-quasicyclic groups via left F{\o}lner sequences of non-empty finite subsets of it, giving a sufficient condition in order a subset of a p-quasicyclic group to contain arbitrary long arithmetic progressions. Finally, we introduce the notion of a dynamical system over p-quasicyclic groups extending the classical notion of a topological dynamical system and we prove (multiple) recurrent results for the p-quasicyclic groups. In particular, we prove recurrent results analogous to Furstenberg-Weiss type theorems for classical systems., Comment: 26 pages. arXiv admin note: substantial text overlap with arXiv:1101.3226 This paper has been withdrawn by the author for further research

Starting with a combinatorial partition theorem for words over an infinite alphabet dominated by a fixed sequence, established recently by the authors, we prove recurrence results for topological dynamical systems indexed by such words. In this way we extend the classical theory developed by Furstenberg and Weiss of dynamical systems indexed by the natural numbers to systems indexed by words. Moreover, applying this theory to topological systems indexed by semigroups that can be represented as words we get analogous recurrence results for such systems., Comment: 20 pages

We introduce the notion of a rational dynamical system extending the classical notion of a topological dynamical system and we prove (multiple) recurrence results for such systems via a partition theorem for the rational numbers proved by Farmaki and the author. In particular, we extend classical recurrence results developed by Furstenberg and Weiss. Also, we give some applications of these topological recurrence results to topology, to combinatorics, to diophantine approximations and to number theory., Comment: 19 pages

Ramsey theory for words over a finite alphabet was unified in the work of Carlson and Furstenberg-Katznelson. Carlson, in the same work, outlined a method to extend the theory for words over an infinite alphabet, but subject to a fixed dominating principle, proving in particular an Ellentuck version, and a corresponding Ramsey theorem for k=1. In the present work we develop in a systematic way a Ramsey theory for words (in fact for {\omega}-Z*-located words) over a doubly infinite alphabet extending Carlson's approach (to countable ordinals and Schreier-type families), and we apply this theory, exploiting the Budak-Isik-Pym representation, to obtain a partition theory for the set of rational numbers. Furthermore, we show that the theory can be used to obtain partition theorems for arbitrary semigroups, stronger than known ones.