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Article Contents

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

Special invited paper

• 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.

Mathematics Subject Classification: Primary: 37A30; Secondary: 37A45, 28D05.

 Citation:

•  [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. [2] V. Bergelson, Weakly mixing PET, Ergodic Theory Dynam. Systems, 7 (1987), 337-349. [3] V. Bergelson, B. Host, R. McCutcheon and F. Parreau, Aspects of uniformity in recurrence, Colloq. Math., 84/85 (2000), 549-576. [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. [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. [6] R. Bowen, Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms, Lecture Notes in Math., 470, Springer-Verlag, Berlin, 1975. [7] R. C. Bradley,  Introduction to Strong Mixing Conditions, Kendrick Press, Heber City, 2007. [8] T. Eisner,  B. Farkas,  M. Haase and  R. Nagel,  Operator Theoretic Aspects of Ergodic Theory, Springer, Cham, 2015. [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. [10] H. Furstenberg,  Recurrence in Ergodic Theory and Combinatorial Number Theory, Princeton Univ. Press, Princeton NJ, 1981. [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. [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. [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. [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. [15] P. R. Halmos,  Lectures on Ergodic Theory, Chelsea Publishing Co., New York, 1965. [16] P. R. Halmos,  Introduction to Hilbert Space and the Theory of Spectral Multiplicity, AMS Chelsea Publishing, Providence, RI, 1998. [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. [18] B. Host and B. Kra, Nonconventional ergodic averages and nilmanifolds, Ann. Math., 161 (2005), 397-488.  doi: 10.4007/annals.2005.161.397. [19] J. Konieczny, Weakly mixing sets of integers and polynomial equations, Quarterly J. Math, 68 (2017), 141-159. [20] A. Leibman, Convergence of multiple ergodic averages along polynomials of several variables, Israel J. Math., 146 (2005), 303-315.  doi: 10.1007/BF02773538. [21] W. Rudin,  Functional Analysis, McGraw-Hill, Inc., New York, 1991. [22] P. Walters,  An Introduction to Ergodic Theory, Springer-Verlag, New York-Berlin, 1982.