# American Institute of Mathematical Sciences

June  2018, 38(6): 2687-2716. doi: 10.3934/dcds.2018113

## Ergodic theorems for nonconventional arrays and an extension of the Szemerédi theorem

 Institute of Mathematics, Hebrew University, Jerusalem 91904, Israel

Special invited paper

Received  July 2017 Revised  November 2017 Published  April 2018

The paper is primarily concerned with the asymptotic behavior as $N\to∞$ of averages of nonconventional arrays having the form ${N^{ - 1}}\sum\limits_{n = 1}^N {\prod\limits_{j = 1}^\ell {{T^{{P_j}(n,N)}}} } {f_j}$ where $f_j$'s are bounded measurable functions, $T$ is an invertible measure preserving transformation and $P_j$'s are polynomials of $n$ and $N$ taking on integer values on integers. It turns out that when $T$ is weakly mixing and $P_j(n, N) = p_jn+q_jN$ are linear or, more generally, have the form $P_j(n, N) = P_j(n)+Q_j(N)$ for some integer valued polynomials $P_j$ and $Q_j$ then the above averages converge in $L^2$ but for general polynomials $P_j$ of both $n$ and $N$ the $L^2$ convergence can be ensured even in the "conventional" case $\ell = 1$ only when $T$ is strongly mixing while for $\ell>1$ strong $2\ell$-mixing should be assumed. Studying also weakly mixing and compact extensions and relying on Furstenberg's structure theorem we derive an extension of Szemerédi's theorem saying that for any subset of integers $\Lambda$ with positive upper density there exists a subset ${\cal N}_\Lambda$ of positive integers having uniformly bounded gaps such that for $N∈{\cal N}_\Lambda$ and at least $\varepsilon N, \, \varepsilon >0$ of $n$'s all numbers $p_jn+q_jN, \, j = 1, ..., \ell,$ belong to $\Lambda$. We obtain also a version of these results for several commuting transformations which yields a corresponding extension of the multidimensional Szemerédi theorem.

Citation: Yuri Kifer. Ergodic theorems for nonconventional arrays and an extension of the Szemerédi theorem. Discrete & Continuous Dynamical Systems, 2018, 38 (6) : 2687-2716. doi: 10.3934/dcds.2018113
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##### References:
 [1] T. Austin, Non-conventional ergodic averages for several commuting actions of an amenable group, J. D'Analyse Math., 130 (2016), 243-274.  doi: 10.1007/s11854-016-0036-6.  Google Scholar [2] V. Bergelson, Weakly mixing PET, Ergodic Theory Dynam. Systems, 7 (1987), 337-349.   Google Scholar [3] V. Bergelson, B. Host, R. McCutcheon and F. Parreau, Aspects of uniformity in recurrence, Colloq. Math., 84/85 (2000), 549-576.   Google Scholar [4] 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.  Google Scholar [5] V. Bergelson, A. Leibman and E. Lesigne, Intersective polynomials and polynomial Szemerédi theorem, Adv. Math., 219 (2008), 369-388.  doi: 10.1016/j.aim.2008.05.008.  Google Scholar [6] R. Bowen, Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms, Lecture Notes in Math., 470, Springer-Verlag, Berlin, 1975.  Google Scholar [7] R. C. Bradley, Introduction to Strong Mixing Conditions, Kendrick Press, Heber City, 2007.   Google Scholar [8] T. Eisner, B. Farkas, M. Haase and R. Nagel, Operator Theoretic Aspects of Ergodic Theory, Springer, Cham, 2015.   Google Scholar [9] H. Furstenberg, Ergodic behavior of diagonal measures and a theorem of Szemerédi on arithmetic progressions, J. d'Analyse Math., 31 (1977), 204-256.  doi: 10.1007/BF02813304.  Google Scholar [10] H. Furstenberg, Recurrence in Ergodic Theory and Combinatorial Number Theory, Princeton Univ. Press, Princeton NJ, 1981.   Google Scholar [11] H. Furstenberg, Nonconventional ergodic averages, in The Legacy of John Von Neumann, Proc. Symp. Pure Math. 50 (1990), 43-56, Amer. Math. Soc., Providence, RI.  Google Scholar [12] H. Furstenberg and Y. Katznelson, An ergodic Szemerédi theorem for commuting transformations, J. d'Analyse Math., 34 (1978), 275-291.  doi: 10.1007/BF02790016.  Google Scholar [13] H. Furstenberg, Y. Katznelson and D. Ornstein, The ergodic theoretical proof of Szemerédi's theorem, Bull. Amer. Math. Soc., 7 (1982), 527-552.  doi: 10.1090/S0273-0979-1982-15052-2.  Google Scholar [14] B. Green and T. Tao, The primes contain arbitrarily long arithmetic progressions, Ann. Math., 167 (2008), 481-547.  doi: 10.4007/annals.2008.167.481.  Google Scholar [15] P. R. Halmos, Lectures on Ergodic Theory, Chelsea Publishing Co., New York, 1965.   Google Scholar [16] P. R. Halmos, Introduction to Hilbert Space and the Theory of Spectral Multiplicity, AMS Chelsea Publishing, Providence, RI, 1998.   Google Scholar [17] L. Heinrich, Mixing properties and central limit theorem for a class of non-identical piecewise monotonic $C^2$-transformations, Mathematische Nachricht, 181 (1996), 185-2014.   Google Scholar [18] B. Host and B. Kra, Nonconventional ergodic averages and nilmanifolds, Ann. Math., 161 (2005), 397-488.  doi: 10.4007/annals.2005.161.397.  Google Scholar [19] J. Konieczny, Weakly mixing sets of integers and polynomial equations, Quarterly J. Math, 68 (2017), 141-159.   Google Scholar [20] A. Leibman, Convergence of multiple ergodic averages along polynomials of several variables, Israel J. Math., 146 (2005), 303-315.  doi: 10.1007/BF02773538.  Google Scholar [21] W. Rudin, Functional Analysis, McGraw-Hill, Inc., New York, 1991.   Google Scholar [22] P. Walters, An Introduction to Ergodic Theory, Springer-Verlag, New York-Berlin, 1982.   Google Scholar
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