Skip to main content Accessibility help
×
Home

Optimal lower bounds for multiple recurrence

  • SEBASTIÁN DONOSO (a1), ANH NGOC LE (a2), JOEL MOREIRA (a2) and WENBO SUN (a3)

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.

Copyright

References

Hide All
[1] Behrend, F. A.. On sets of integers which contain no three terms in arithmetical progression. Proc. Natl Acad. Sci. USA 32 (1946), 331332.
[2] Bergelson, V.. Weakly mixing PET. Ergod. Th. & Dynam. Sys. 7(3) (1987), 337349.
[3] 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), 13751416.
[4] Bergelson, V., Host, B. and Kra, B.. Multiple recurrence and nilsequences. Invent. Math. 160(2) (2005), 261303. With an appendix by I. Ruzsa.
[5] Bergelson, V., Moreira, J. and Richter, F. K.. Single and multiple recurrence along non-polynomial sequences. Preprint, 2017, arXiv:1711.05729.
[6] Chu, Q.. Multiple recurrence for two commuting transformations. Ergod. Th. & Dynam. Sys. 31(3) (2011), 771792.
[7] Donoso, S. and Sun, W.. Quantitative multiple recurrence for two and three transformations. Israel J. Math. 226(1) (2018), 7185.
[8] Frantzikinakis, N.. The structure of strongly stationary systems. J. Anal. Math. 93 (2004), 359388.
[9] Frantzikinakis, N.. Multiple ergodic averages for three polynomials and applications. Trans. Amer. Math. Soc. 360(10) (2008), 54355475.
[10] Frantzikinakis, N.. Equidistribution of sparse sequences on nilmanifolds. J. Anal. Math. 109 (2009), 353395.
[11] Frantzikinakis, N.. Multiple recurrence and convergence for Hardy sequences of polynomial growth. J. Anal. Math. 112 (2010), 79135.
[12] Frantzikinakis, N.. A multidimensional Szemerédi theorem for Hardy sequences of different growth. Trans. Amer. Math. Soc. 367(8) (2015), 56535692.
[13] Frantzikinakis, N., Host, B. and Kra, B.. Multiple recurrence and convergence for sequences related to the prime numbers. J. Reine Angew. Math. 611 (2007), 131144.
[14] Frantzikinakis, N., Host, B. and Kra, B.. The polynomial multidimensional Szemerédi theorem along shifted primes. Israel J. Math. 194(1) (2013), 331348.
[15] Frantzikinakis, N. and Kra, B.. Polynomial averages converge to the product of integrals. Israel J. Math. 148 (2005), 267276.
[16] Frantzikinakis, N. and Kra, B.. Ergodic averages for independent polynomials and applications. J. Lond. Math. Soc. (2) 74(1) (2006), 131142.
[17] Frantzikinakis, N. and Wierdl, M.. A Hardy field extension of Szemerédi’s theorem. Adv. Math. 222(1) (2009), 143.
[18] Furstenberg, H.. Ergodic behavior of diagonal measures and a theorem of Szemerédi on arithmetic progressions. J. Anal. Math. 31 (1977), 204256.
[19] Host, B. and Kra, B.. An odd Furstenberg-Szemerédi theorem and quasi-affine systems. J. Anal. Math. 86 (2002), 183220.
[20] Host, B. and Kra, B.. Nonconventional ergodic averages and nilmanifolds. Ann. of Math. (2) 161(1) (2005), 397488.
[21] Host, B. and Kra, B.. Nilpotent Structures in Ergodic Theory (Mathematical Surveys and Monographs, 236) . American Mathematical Society, Providence, RI, 2018.
[22] Khintchine, A.. Korrelationstheorie der stationären stochastischen Prozesse. Math. Ann. 109(1) (1934), 604615.
[23] 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.
[24] Leibman, A.. Multiple polynomial correlation sequences and nilsequences. Ergod. Th. & Dynam. Sys. 30(3) (2010), 841854.
[25] Leibman, A.. Orbit of the diagonal in the power of a nilmanifold. Trans. Amer. Math. Soc. 362(3) (2010), 16191658.
[26] 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), 10421070.
[27] Ratner, M.. Raghunathan’s topological conjecture and distributions of unipotent flows. Duke Math. J. 63(1) (1991), 235280.
[28] Shiu, D.. Strings of congruent primes. J. Lond. Math. Soc. (2) 61(2) (2000), 359373.
[29] Wooley, T. and Ziegler, T.. Multiple recurrence and convergence along the primes. Amer. J. Math. 134(6) (2012), 17051732.
[30] Ziegler, T.. A non-conventional ergodic theorem for a nilsystem. Ergod. Th. & Dynam. Sys. 25(4) (2005), 13571370.
[31] Ziegler, T.. Universal characteristic factors and Furstenberg averages. J. Amer. Math. Soc. 20(1) (2007), 5397.

Keywords

MSC classification

Optimal lower bounds for multiple recurrence

  • SEBASTIÁN DONOSO (a1), ANH NGOC LE (a2), JOEL MOREIRA (a2) and WENBO SUN (a3)

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed