# American Institute of Mathematical Sciences

June  2021, 41(6): 2809-2828. doi: 10.3934/dcds.2020386

## Properties of multicorrelation sequences and large returns under some ergodicity assumptions

 Department of Mathematics, Ohio State University, Columbus, OH 43210, USA

Received  June 2020 Revised  September 2020 Published  June 2021 Early access  November 2020

We prove that given a measure preserving system $(X,\mathcal{B},\mu,T_1,\dots, T_d)$ with commuting, ergodic transformations $T_i$ such that $T_iT_j^{-1}$ are ergodic for all $i \neq j$, the multicorrelation sequence $a(n) = \int_X f_0 \cdot T_1^nf_1 \cdot \dotso \cdot T_d^n f_d \ d\mu$ can be decomposed as $a(n) = a_{ \rm{st}}(n)+a_{ \rm{er}}(n)$, where $a_{ \rm{st}}$ is a uniform limit of $d$-step nilsequences and $a_{ \rm{er}}$ is a nullsequence (that is, $\lim_{N-M \to \infty} \frac{1}{N-M} \sum_{n = M}^{N-1} |a_{ \rm{er}}|^2 = 0$). Under some additional ergodicity conditions on $T_1,\dots,T_d$ we also establish a similar decomposition for polynomial multicorrelation sequences of the form $a(n) = \int_X f_0 \cdot \prod_{i = 1}^dT_i^{p_{i,1}(n)}f_1\cdot\dotso \cdot \prod_{i = 1}^dT_i^{p_{i,k}(n)}f_k \ d\mu$, where each $p_{i,k}: {\mathbb{Z}} \rightarrow {\mathbb{Z}}$ is a polynomial map. We also show, for $d = 2$, that if $T_1, T_2, T_1T_2^{-1}$ are invertible and ergodic, we have large triple intersections: for all $\varepsilon>0$ and all $A \in \mathcal{B}$, the set $\{n \in {\mathbb{Z}} : \mu(A \cap T_1^{-n}A \cap T_2^{-n}A)>\mu(A)^3-\varepsilon\}$ is syndetic. Moreover, we show that if $T_1, T_2, T_1T_2^{-1}$ are totally ergodic, and we denote by $p_n$ the $n$-th prime, the set $\{n \in \mathbb{N} : \mu(A \cap T_1^{-(p_n-1)}A \cap T_2^{-(p_n-1)}A)>\mu(A)^3-\varepsilon\}$ has positive lower density.

Citation: Andreu Ferré Moragues. Properties of multicorrelation sequences and large returns under some ergodicity assumptions. Discrete and Continuous Dynamical Systems, 2021, 41 (6) : 2809-2828. doi: 10.3934/dcds.2020386
##### References:
 [1] V. Bergelson, Weakly mixing PET, Ergodic Theory Dynam. Systems, 7 (1987), 337-349.  doi: 10.1017/S0143385700004090. [2] V. Bergelson, B. Host and B. Kra, Multiple recurrence and nilsequences. With an appendix by Imre Rusza, Invent. Math., 160 (2005), 261-303.  doi: 10.1007/s00222-004-0428-6. [3] V. Bergelson and A. Leibman, Polynomial extensions of van der Waerden's and Szemerédi's theorems, J. Amer. Math. Soc., 9 (1996), 725-753.  doi: 10.1090/S0894-0347-96-00194-4. [4] V. Bergelson and A. Leibman, Cubic averages and large intersections, Recent Trends in Ergodic Theory and Dynamical Systems, Contemp. Math., 631, Amer. Math. Soc., Providence, RI, 5–19. doi: 10.1090/conm/631/12592. [5] V. Bergelson, T. Tao and T. Ziegler, Multiple recurrence and convergence results associated to $\Bbb {F}_p^\omega$-actions, J. Anal. Math., 127 (2015), 329-378.  doi: 10.1007/s11854-015-0033-1. [6] Q. Chu, Multiple recurrence for two commuting transformations, Ergodic Theory Dynam. Systems, 31 (2011), 771-792.  doi: 10.1017/S0143385710000258. [7] S. Donoso, J. Moreira, A. N. Le and W. Sun, Optimal lower bounds for multiple recurrence, Ergodic Theory and Dynamical Systems, (2019), 1–29. doi: 10.1017/etds.2019.72. [8] S. Donoso and W. Sun, Quantitative multiple recurrence for two and three transformations, Israel J. Math., 226 (2018), 71-85.  doi: 10.1007/s11856-018-1690-4. [9] N. Frantzikinakis, Multiple correlation sequences and nilsequences, Invent. Math., 202 (2015), 875-892.  doi: 10.1007/s00222-015-0579-7. [10] N. Frantzikinakis, Multiple ergodic averages for three polynomials and applications, Trans. Amer. Math. Soc., 360 (2008), 5435-5475.  doi: 10.1090/S0002-9947-08-04591-1. [11] N. Frantzikinakis and B. Host, Weighted multiple ergodic averages and correlation sequences, Ergodic Theory Dynam. Systems, 38 (2018), 81-142.  doi: 10.1017/etds.2016.19. [12] N. Frantzikinakis, B. Host and B. Kra, The polynomial multidimensional Szemerédi theorem along shifted primes, Israel J. Math., 194 (2013), 331-348.  doi: 10.1007/s11856-012-0132-y. [13] N. Frantzikinakis and B. Kra, Convergence of multiple ergodic averages for some commuting transformations, Ergodic Theory Dynam. Systems, 25 (2005), 799-809.  doi: 10.1017/S0143385704000616. [14] W. T. Gowers, A new proof of Szemerédi's theorem, Geom. Funct. Anal., 11 (2001), 465-588.  doi: 10.1007/s00039-001-0332-9. [15] J. T. Griesmer, Ergodic Averages, Correlation Sequences, and Sumsets, Ph. D thesis, The Ohio State University, 2009. [16] B. Host, Ergodic seminorms for commuting transformations and applications, Studia Math., 195 (2009), 31-49.  doi: 10.4064/sm195-1-3. [17] B. Host and B. Kra, Nonconventional ergodic averages and nilmanifolds, Ann. of Math., 161 (2005), 397-488.  doi: 10.4007/annals.2005.161.397. [18] B. Host and B. Kra, Nilpotent Structures in Ergodic Theory, Mathematical Surveys and Monographs, 236, American Mathematical Society, Providence, RI, 2018. doi: 10.1090/surv/236. [19] M. C. R. Johnson, Convergence of polynomial ergodic averages of several variables for some commuting transformations, Illinois J. Math., 53 (2009), 865-882.  doi: 10.1215/ijm/1286212920. [20] A. Khintchine, The method of spectral reduction in classical dynamics, Proceedings of the National Academy of Sciences, 19 (1933), 567-573.  doi: 10.1073/pnas.19.5.567. [21] B. O. Koopman and J. von Neumann, Dynamical systems of continuous spectra, Proceedings of the National Academy of Sciences, 18 (1932), 255-263.  doi: 10.1073/pnas.18.3.255. [22] A. Koutsogiannis, A. Le, J. Moreira, and F. K. Richter, Structure of multicorrelation sequences with integer part polynomial iterates along primes, Proc. Amer. Math. Soc. 149 (2021), no. 1,209–216. [23] A. N. Le, Nilsequences and multiple correlations along subsequences, Ergodic Theory Dynam. Systems, 40 (2020), 1634-1654.  doi: 10.1017/etds.2018.110. [24] A. Leibman, Multiple polynomial correlation sequences and nilsequences, Ergodic Theory Dynam. Systems, 30 (2010), 841-854.  doi: 10.1017/S0143385709000303. [25] A. Leibman, Nilsequences, null-sequences, and multiple correlation sequences, Ergodic Theory Dynam. Systems, 35 (2015), 176-191.  doi: 10.1017/etds.2013.36. [26] P. Walters, An introduction to Ergodic Theory, Graduate Texts in Mathematics, 79, Springer-Verlag, New York, 1982.

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##### References:
 [1] V. Bergelson, Weakly mixing PET, Ergodic Theory Dynam. Systems, 7 (1987), 337-349.  doi: 10.1017/S0143385700004090. [2] V. Bergelson, B. Host and B. Kra, Multiple recurrence and nilsequences. With an appendix by Imre Rusza, Invent. Math., 160 (2005), 261-303.  doi: 10.1007/s00222-004-0428-6. [3] V. Bergelson and A. Leibman, Polynomial extensions of van der Waerden's and Szemerédi's theorems, J. Amer. Math. Soc., 9 (1996), 725-753.  doi: 10.1090/S0894-0347-96-00194-4. [4] V. Bergelson and A. Leibman, Cubic averages and large intersections, Recent Trends in Ergodic Theory and Dynamical Systems, Contemp. Math., 631, Amer. Math. Soc., Providence, RI, 5–19. doi: 10.1090/conm/631/12592. [5] V. Bergelson, T. Tao and T. Ziegler, Multiple recurrence and convergence results associated to $\Bbb {F}_p^\omega$-actions, J. Anal. Math., 127 (2015), 329-378.  doi: 10.1007/s11854-015-0033-1. [6] Q. Chu, Multiple recurrence for two commuting transformations, Ergodic Theory Dynam. Systems, 31 (2011), 771-792.  doi: 10.1017/S0143385710000258. [7] S. Donoso, J. Moreira, A. N. Le and W. Sun, Optimal lower bounds for multiple recurrence, Ergodic Theory and Dynamical Systems, (2019), 1–29. doi: 10.1017/etds.2019.72. [8] S. Donoso and W. Sun, Quantitative multiple recurrence for two and three transformations, Israel J. Math., 226 (2018), 71-85.  doi: 10.1007/s11856-018-1690-4. [9] N. Frantzikinakis, Multiple correlation sequences and nilsequences, Invent. Math., 202 (2015), 875-892.  doi: 10.1007/s00222-015-0579-7. [10] N. Frantzikinakis, Multiple ergodic averages for three polynomials and applications, Trans. Amer. Math. Soc., 360 (2008), 5435-5475.  doi: 10.1090/S0002-9947-08-04591-1. [11] N. Frantzikinakis and B. Host, Weighted multiple ergodic averages and correlation sequences, Ergodic Theory Dynam. Systems, 38 (2018), 81-142.  doi: 10.1017/etds.2016.19. [12] N. Frantzikinakis, B. Host and B. Kra, The polynomial multidimensional Szemerédi theorem along shifted primes, Israel J. Math., 194 (2013), 331-348.  doi: 10.1007/s11856-012-0132-y. [13] N. Frantzikinakis and B. Kra, Convergence of multiple ergodic averages for some commuting transformations, Ergodic Theory Dynam. Systems, 25 (2005), 799-809.  doi: 10.1017/S0143385704000616. [14] W. T. Gowers, A new proof of Szemerédi's theorem, Geom. Funct. Anal., 11 (2001), 465-588.  doi: 10.1007/s00039-001-0332-9. [15] J. T. Griesmer, Ergodic Averages, Correlation Sequences, and Sumsets, Ph. D thesis, The Ohio State University, 2009. [16] B. Host, Ergodic seminorms for commuting transformations and applications, Studia Math., 195 (2009), 31-49.  doi: 10.4064/sm195-1-3. [17] B. Host and B. Kra, Nonconventional ergodic averages and nilmanifolds, Ann. of Math., 161 (2005), 397-488.  doi: 10.4007/annals.2005.161.397. [18] B. Host and B. Kra, Nilpotent Structures in Ergodic Theory, Mathematical Surveys and Monographs, 236, American Mathematical Society, Providence, RI, 2018. doi: 10.1090/surv/236. [19] M. C. R. Johnson, Convergence of polynomial ergodic averages of several variables for some commuting transformations, Illinois J. Math., 53 (2009), 865-882.  doi: 10.1215/ijm/1286212920. [20] A. Khintchine, The method of spectral reduction in classical dynamics, Proceedings of the National Academy of Sciences, 19 (1933), 567-573.  doi: 10.1073/pnas.19.5.567. [21] B. O. Koopman and J. von Neumann, Dynamical systems of continuous spectra, Proceedings of the National Academy of Sciences, 18 (1932), 255-263.  doi: 10.1073/pnas.18.3.255. [22] A. Koutsogiannis, A. Le, J. Moreira, and F. K. Richter, Structure of multicorrelation sequences with integer part polynomial iterates along primes, Proc. Amer. Math. Soc. 149 (2021), no. 1,209–216. [23] A. N. Le, Nilsequences and multiple correlations along subsequences, Ergodic Theory Dynam. Systems, 40 (2020), 1634-1654.  doi: 10.1017/etds.2018.110. [24] A. Leibman, Multiple polynomial correlation sequences and nilsequences, Ergodic Theory Dynam. Systems, 30 (2010), 841-854.  doi: 10.1017/S0143385709000303. [25] A. Leibman, Nilsequences, null-sequences, and multiple correlation sequences, Ergodic Theory Dynam. Systems, 35 (2015), 176-191.  doi: 10.1017/etds.2013.36. [26] P. Walters, An introduction to Ergodic Theory, Graduate Texts in Mathematics, 79, Springer-Verlag, New York, 1982.
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