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Optimal lower bounds for multiple recurrence

Part of: Ergodic theory Probabilistic theory: distribution modulo $1$; metric theory of algorithms

Published online by Cambridge University Press:  07 October 2019

SEBASTIÁN DONOSO Open the ORCID record for SEBASTIÁN DONOSO [Opens in a new window] ,
ANH NGOC LE Open the ORCID record for ANH NGOC LE [Opens in a new window] ,
JOEL MOREIRA Open the ORCID record for JOEL MOREIRA [Opens in a new window]  and
WENBO SUN Open the ORCID record for WENBO SUN [Opens in a new window]
SEBASTIÁN DONOSO
Affiliation:
Instituto de Ciencias de la Ingeniería, Universidad de O’Higgings, Av. Libertador Bernardo O’Higgins 611, Rancagua, Chile email sebastian.donoso@uoh.cl
ANH NGOC LE
Affiliation:
Department of Mathematics, Northwestern University, 2033 Sheridan Road, Evanston, IL60208-2730, USA email anhle@math.northwestern.edu, joel.moreira@northwestern.edu
JOEL MOREIRA
Affiliation:
Department of Mathematics, Northwestern University, 2033 Sheridan Road, Evanston, IL60208-2730, USA email anhle@math.northwestern.edu, joel.moreira@northwestern.edu
WENBO SUN
Affiliation:
Department of Mathematics, The Ohio State University, 231 West 18th Avenue, Columbus, OH43210-1174, USA email sun.1991@osu.edu
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Abstract

Let $(X,{\mathcal{B}},\unicode[STIX]{x1D707},T)$ be an ergodic measure-preserving system, let $A\in {\mathcal{B}}$ and let $\unicode[STIX]{x1D716}>0$. We study the largeness of sets of the form

$$\begin{eqnarray}S=\{n\in \mathbb{N}:\unicode[STIX]{x1D707}(A\cap T^{-f_{1}(n)}A\cap T^{-f_{2}(n)}A\cap \cdots \cap T^{-f_{k}(n)}A)>\unicode[STIX]{x1D707}(A)^{k+1}-\unicode[STIX]{x1D716}\}\end{eqnarray}$$
for various families $\{f_{1},\ldots ,f_{k}\}$ of sequences $f_{i}:\mathbb{N}\rightarrow \mathbb{N}$. For $k\leq 3$ and $f_{i}(n)=if(n)$, we show that $S$ has positive density if $f(n)=q(p_{n})$, where $q\in \mathbb{Z}[x]$ satisfies $q(1)$ or $q(-1)=0$ and $p_{n}$ denotes the $n$th prime; or when $f$ is a certain Hardy field sequence. If $T^{q}$ is ergodic for some $q\in \mathbb{N}$, then, for all $r\in \mathbb{Z}$, $S$ is syndetic if $f(n)=qn+r$. For $f_{i}(n)=a_{i}n$, where $a_{i}$ are distinct integers, we show that $S$ can be empty for $k\geq 4$, and, for $k=3$, we found an interesting relation between the largeness of $S$ and the abundance of solutions to certain linear equations in sparse sets of integers. We also provide some partial results when the $f_{i}$ are distinct polynomials.

Keywords

optimal multiple recurrencesystems of linear equationsnilsystems

MSC classification

Primary: 37A45: Relations with number theory and harmonic analysis
Secondary: 37A30: Ergodic theorems, spectral theory, Markov operators 11K36: Well-distributed sequences and other variations
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 41 , Issue 2 , February 2021 , pp. 379 - 407
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Copyright
© Cambridge University Press, 2019

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References

Behrend, F. A.. On sets of integers which contain no three terms in arithmetical progression. Proc. Natl Acad. Sci. USA 32 (1946), 331–332.CrossRefGoogle ScholarPubMed
Bergelson, V.. Weakly mixing PET. Ergod. Th. & Dynam. Sys. 7(3) (1987), 337–349.CrossRefGoogle Scholar
Bergelson, V. and Håland Knutson, I. J.. Weak mixing implies weak mixing of higher orders along tempered functions. Ergod. Th. & Dynam. Sys. 29(5) (2009), 1375–1416.CrossRefGoogle Scholar
Bergelson, V., Host, B. and Kra, B.. Multiple recurrence and nilsequences. Invent. Math. 160(2) (2005), 261–303. With an appendix by I. Ruzsa.CrossRefGoogle Scholar
Bergelson, V., Moreira, J. and Richter, F. K.. Single and multiple recurrence along non-polynomial sequences. Preprint, 2017, arXiv:1711.05729.Google Scholar
Chu, Q.. Multiple recurrence for two commuting transformations. Ergod. Th. & Dynam. Sys. 31(3) (2011), 771–792.10.1017/S0143385710000258CrossRefGoogle Scholar
Donoso, S. and Sun, W.. Quantitative multiple recurrence for two and three transformations. Israel J. Math. 226(1) (2018), 71–85.CrossRefGoogle Scholar
Frantzikinakis, N.. The structure of strongly stationary systems. J. Anal. Math. 93 (2004), 359–388.CrossRefGoogle Scholar
Frantzikinakis, N.. Multiple ergodic averages for three polynomials and applications. Trans. Amer. Math. Soc. 360(10) (2008), 5435–5475.CrossRefGoogle Scholar
Frantzikinakis, N.. Equidistribution of sparse sequences on nilmanifolds. J. Anal. Math. 109 (2009), 353–395.CrossRefGoogle Scholar
Frantzikinakis, N.. Multiple recurrence and convergence for Hardy sequences of polynomial growth. J. Anal. Math. 112 (2010), 79–135.CrossRefGoogle Scholar
Frantzikinakis, N.. A multidimensional Szemerédi theorem for Hardy sequences of different growth. Trans. Amer. Math. Soc. 367(8) (2015), 5653–5692.CrossRefGoogle Scholar
Frantzikinakis, N., Host, B. and Kra, B.. Multiple recurrence and convergence for sequences related to the prime numbers. J. Reine Angew. Math. 611 (2007), 131–144.Google Scholar
Frantzikinakis, N., Host, B. and Kra, B.. The polynomial multidimensional Szemerédi theorem along shifted primes. Israel J. Math. 194(1) (2013), 331–348.CrossRefGoogle Scholar
Frantzikinakis, N. and Kra, B.. Polynomial averages converge to the product of integrals. Israel J. Math. 148 (2005), 267–276.CrossRefGoogle Scholar
Frantzikinakis, N. and Kra, B.. Ergodic averages for independent polynomials and applications. J. Lond. Math. Soc. (2) 74(1) (2006), 131–142.CrossRefGoogle Scholar
Frantzikinakis, N. and Wierdl, M.. A Hardy field extension of Szemerédi’s theorem. Adv. Math. 222(1) (2009), 1–43.CrossRefGoogle Scholar
Furstenberg, H.. Ergodic behavior of diagonal measures and a theorem of Szemerédi on arithmetic progressions. J. Anal. Math. 31 (1977), 204–256.CrossRefGoogle Scholar
Host, B. and Kra, B.. An odd Furstenberg-Szemerédi theorem and quasi-affine systems. J. Anal. Math. 86 (2002), 183–220.CrossRefGoogle Scholar
Host, B. and Kra, B.. Nonconventional ergodic averages and nilmanifolds. Ann. of Math. (2) 161(1) (2005), 397–488.CrossRefGoogle Scholar
Host, B. and Kra, B.. Nilpotent Structures in Ergodic Theory (Mathematical Surveys and Monographs, 236) . American Mathematical Society, Providence, RI, 2018.CrossRefGoogle Scholar
Khintchine, A.. Korrelationstheorie der stationären stochastischen Prozesse. Math. Ann. 109(1) (1934), 604–615.CrossRefGoogle Scholar
Le, A. N.. Nilsequences and multiple correlations along subsequences. Ergod. Th. & Dynam. Sys., to appear. Preprint, 2018, https://doi.org/10.1017/etds.2018.110.CrossRefGoogle Scholar
Leibman, A.. Multiple polynomial correlation sequences and nilsequences. Ergod. Th. & Dynam. Sys. 30(3) (2010), 841–854.CrossRefGoogle Scholar
Leibman, A.. Orbit of the diagonal in the power of a nilmanifold. Trans. Amer. Math. Soc. 362(3) (2010), 1619–1658.CrossRefGoogle Scholar
Moreira, J. and Richter, F.. A spectral refinement of the Bergelson–Host–Kra decomposition and new multiple ergodic theorems. Ergod. Th. & Dynam. Sys. 39(4) (2019), 1042–1070.CrossRefGoogle Scholar
Ratner, M.. Raghunathan’s topological conjecture and distributions of unipotent flows. Duke Math. J. 63(1) (1991), 235–280.CrossRefGoogle Scholar
Shiu, D.. Strings of congruent primes. J. Lond. Math. Soc. (2) 61(2) (2000), 359–373.CrossRefGoogle Scholar
Wooley, T. and Ziegler, T.. Multiple recurrence and convergence along the primes. Amer. J. Math. 134(6) (2012), 1705–1732.CrossRefGoogle Scholar
Ziegler, T.. A non-conventional ergodic theorem for a nilsystem. Ergod. Th. & Dynam. Sys. 25(4) (2005), 1357–1370.CrossRefGoogle Scholar
Ziegler, T.. Universal characteristic factors and Furstenberg averages. J. Amer. Math. Soc. 20(1) (2007), 53–97.CrossRefGoogle Scholar