doi: 10.3934/dcds.2022067
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Multiple ergodic averages for variable polynomials

Aristotle University of Thessaloniki, Department of Mathematics, Thessaloniki, 54124, Greece

Dedicated to the loving memory of Aris Deligiannis, a great mentor

Received  November 2021 Revised  March 2022 Early access May 2022

In this paper we study multiple ergodic averages for "good" variable polynomials. In particular, under an additional assumption, we show that these averages converge to the expected limit, making progress related to an open problem posted by Frantzikinakis ([13,Problem 10]). These general convergence results imply several variable extensions of classical recurrence, combinatorial and number theoretical results which are presented as well.

Citation: Andreas Koutsogiannis. Multiple ergodic averages for variable polynomials. Discrete and Continuous Dynamical Systems, doi: 10.3934/dcds.2022067
References:
[1]

V. Bergelson, Ergodic Ramsey theory, Logic and Combinatorics (Arcata, Calif., 1985), Contemp. Math. Amer. Math. Soc., Providence, RI, 65 (1987), 63–87. doi: 10.1090/conm/065/891243.

[2]

V. Bergelson, Weakly mixing PET, Ergodic Theory Dynam. Systems, 7 (1987), 337-349.  doi: 10.1017/S0143385700004090.

[3]

V. Bergelson and I. Håland-Knutson, Weakly mixing implies mixing of higher orders along tempered functions, Ergodic Theory Dynam. Systems, 29 (2009), 1375-1416.  doi: 10.1017/S0143385708000862.

[4]

V. Bergelson and A. Leibman, A nilpotent Roth theorem, Invent. Math., 147 (2002), 429-470.  doi: 10.1007/s002220100179.

[5]

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.

[6]

Q. ChuN. Frantzikinakis and B. Host, Ergodic averages of commuting transformations with distinct degree polynomial iterates, Proc. Lond. Math. Soc., 102 (2011), 801-842.  doi: 10.1112/plms/pdq037.

[7]

S. Donoso, A. Ferré Moragues, A. Koutsogiannis and W. Sun, Decomposition of multicorrelation sequences and joint ergodicity, preprint, 2021, arXiv: 2106.01058.

[8]

S. DonosoA. Koutsogiannis and W. Sun, Pointwise multiple averages for sublinear functions, Ergodic Theory Dynam. Systems, 40 (2020), 1594-1618.  doi: 10.1017/etds.2018.118.

[9]

S. DonosoA. Koutsogiannis and W. Sun, Seminorms for multiple averages along polynomials and applications to joint ergodicity, J. d'Analyse Math., (2021).  doi: 10.1007/s11854-021-0186-z.

[10]

N. Frantzikinakis, A multidimensional Szemerédi theorem for Hardy sequences of different growth, Trans. Amer. Math. Soc., 367 (2015), 5653-5692.  doi: 10.1090/S0002-9947-2014-06275-2.

[11]

N. Frantzikinakis, Equidistribution of sparse sequences on nilmanifolds, J. Anal. Math., 109 (2009), 353-395.  doi: 10.1007/s11854-009-0035-y.

[12]

N. Frantzikinakis, Multiple recurrence and convergence for Hardy sequences of polynomial growth, J. Anal. Math., 112 (2010), 79-135.  doi: 10.1007/s11854-010-0026-z.

[13]

N. Frantzikinakis, Some open problems on multiple ergodic averages, Bull. Hellenic Math. Soc., 60 (2016), 41-90. 

[14]

N. Frantzikinakis, An averaged Chowla and Elliott conjecture along independent polynomials, Int. Math. Res. Not. IMRN, 2018 (2018), 3721-3743.  doi: 10.1093/imrn/rnx002.

[15]

N. FrantzikinakisB. 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.

[16]

H. Furstenberg, Ergodic behavior of diagonal measures and a theorem of Szemerédi on arithmetic progressions, J. Analyse Math., 31 (1977), 204-256.  doi: 10.1007/BF02813304.

[17]

B. Green and T. Tao, The quantitative behaviour of polynomial orbits on nilmanifolds, Ann. of Math., 175 (2012), 465-540.  doi: 10.4007/annals.2012.175.2.2.

[18]

G. H. Hardy, Proc. of the London Math. Society, Proceedings of the London Mathematical Society, s2-10 (1912), 54-90.  doi: 10.1112/plms/s2-10.1.54.

[19]

B. Host and B. Kra, Nonconventional ergodic averages and nilmanifolds, Annals of Math., 161 (2005), 397-488.  doi: 10.4007/annals.2005.161.397.

[20]

D. Karageorgos and A. Koutsogiannis, Integer part independent polynomial averages and applications along primes, Studia Math., 249 (2019), 233-257.  doi: 10.4064/sm171102-18-9.

[21]

Y. Kifer, Ergodic theorems for nonconventional arrays and an extension of the Szemerédi theorem, Discrete Contin. Dyn. Syst., 38 (2018), 2687-2716.  doi: 10.3934/dcds.2018113.

[22]

A. Koutsogiannis, Closest integer polynomial multiple recurrence along shifted primes, Ergodic Theory Dynam. Systems, 38 (2018), 666-685.  doi: 10.1017/etds.2016.40.

[23]

A. Koutsogiannis, Integer part polynomial correlation sequences, Ergodic Theory Dynam. Systems, 38 (2018), 1525-1542.  doi: 10.1017/etds.2016.67.

[24]

A Koutsogiannis, Multiple ergodic averages for tempered functions, Discrete Contin. Dyn. Syst., 41 (2021), 1177-1205.  doi: 10.3934/dcds.2020314.

[25]

A. Leibman, Pointwise Convergence of ergodic averages for polynomial sequences of translations on a nilmanifold, Ergodic Theory Dynam. Systems, 25 (2005), 201-213.  doi: 10.1017/S0143385704000215.

[26]

M. Ratner, Raghunatan's topological conjecture and distribution of unipotent flows, Duke Math. J., 63 (1991), 235-280.  doi: 10.1215/S0012-7094-91-06311-8.

[27]

M. Walsh, Norm convergence of nilpotent ergodic averages, Annals of Math., 175 (2012), 1667-1688.  doi: 10.4007/annals.2012.175.3.15.

[28]

T. Ziegler, Universal characteristic factors and Furstenberg averages, J. Amer. Math. Soc., 20 (2007), 53-97.  doi: 10.1090/S0894-0347-06-00532-7.

show all references

References:
[1]

V. Bergelson, Ergodic Ramsey theory, Logic and Combinatorics (Arcata, Calif., 1985), Contemp. Math. Amer. Math. Soc., Providence, RI, 65 (1987), 63–87. doi: 10.1090/conm/065/891243.

[2]

V. Bergelson, Weakly mixing PET, Ergodic Theory Dynam. Systems, 7 (1987), 337-349.  doi: 10.1017/S0143385700004090.

[3]

V. Bergelson and I. Håland-Knutson, Weakly mixing implies mixing of higher orders along tempered functions, Ergodic Theory Dynam. Systems, 29 (2009), 1375-1416.  doi: 10.1017/S0143385708000862.

[4]

V. Bergelson and A. Leibman, A nilpotent Roth theorem, Invent. Math., 147 (2002), 429-470.  doi: 10.1007/s002220100179.

[5]

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.

[6]

Q. ChuN. Frantzikinakis and B. Host, Ergodic averages of commuting transformations with distinct degree polynomial iterates, Proc. Lond. Math. Soc., 102 (2011), 801-842.  doi: 10.1112/plms/pdq037.

[7]

S. Donoso, A. Ferré Moragues, A. Koutsogiannis and W. Sun, Decomposition of multicorrelation sequences and joint ergodicity, preprint, 2021, arXiv: 2106.01058.

[8]

S. DonosoA. Koutsogiannis and W. Sun, Pointwise multiple averages for sublinear functions, Ergodic Theory Dynam. Systems, 40 (2020), 1594-1618.  doi: 10.1017/etds.2018.118.

[9]

S. DonosoA. Koutsogiannis and W. Sun, Seminorms for multiple averages along polynomials and applications to joint ergodicity, J. d'Analyse Math., (2021).  doi: 10.1007/s11854-021-0186-z.

[10]

N. Frantzikinakis, A multidimensional Szemerédi theorem for Hardy sequences of different growth, Trans. Amer. Math. Soc., 367 (2015), 5653-5692.  doi: 10.1090/S0002-9947-2014-06275-2.

[11]

N. Frantzikinakis, Equidistribution of sparse sequences on nilmanifolds, J. Anal. Math., 109 (2009), 353-395.  doi: 10.1007/s11854-009-0035-y.

[12]

N. Frantzikinakis, Multiple recurrence and convergence for Hardy sequences of polynomial growth, J. Anal. Math., 112 (2010), 79-135.  doi: 10.1007/s11854-010-0026-z.

[13]

N. Frantzikinakis, Some open problems on multiple ergodic averages, Bull. Hellenic Math. Soc., 60 (2016), 41-90. 

[14]

N. Frantzikinakis, An averaged Chowla and Elliott conjecture along independent polynomials, Int. Math. Res. Not. IMRN, 2018 (2018), 3721-3743.  doi: 10.1093/imrn/rnx002.

[15]

N. FrantzikinakisB. 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.

[16]

H. Furstenberg, Ergodic behavior of diagonal measures and a theorem of Szemerédi on arithmetic progressions, J. Analyse Math., 31 (1977), 204-256.  doi: 10.1007/BF02813304.

[17]

B. Green and T. Tao, The quantitative behaviour of polynomial orbits on nilmanifolds, Ann. of Math., 175 (2012), 465-540.  doi: 10.4007/annals.2012.175.2.2.

[18]

G. H. Hardy, Proc. of the London Math. Society, Proceedings of the London Mathematical Society, s2-10 (1912), 54-90.  doi: 10.1112/plms/s2-10.1.54.

[19]

B. Host and B. Kra, Nonconventional ergodic averages and nilmanifolds, Annals of Math., 161 (2005), 397-488.  doi: 10.4007/annals.2005.161.397.

[20]

D. Karageorgos and A. Koutsogiannis, Integer part independent polynomial averages and applications along primes, Studia Math., 249 (2019), 233-257.  doi: 10.4064/sm171102-18-9.

[21]

Y. Kifer, Ergodic theorems for nonconventional arrays and an extension of the Szemerédi theorem, Discrete Contin. Dyn. Syst., 38 (2018), 2687-2716.  doi: 10.3934/dcds.2018113.

[22]

A. Koutsogiannis, Closest integer polynomial multiple recurrence along shifted primes, Ergodic Theory Dynam. Systems, 38 (2018), 666-685.  doi: 10.1017/etds.2016.40.

[23]

A. Koutsogiannis, Integer part polynomial correlation sequences, Ergodic Theory Dynam. Systems, 38 (2018), 1525-1542.  doi: 10.1017/etds.2016.67.

[24]

A Koutsogiannis, Multiple ergodic averages for tempered functions, Discrete Contin. Dyn. Syst., 41 (2021), 1177-1205.  doi: 10.3934/dcds.2020314.

[25]

A. Leibman, Pointwise Convergence of ergodic averages for polynomial sequences of translations on a nilmanifold, Ergodic Theory Dynam. Systems, 25 (2005), 201-213.  doi: 10.1017/S0143385704000215.

[26]

M. Ratner, Raghunatan's topological conjecture and distribution of unipotent flows, Duke Math. J., 63 (1991), 235-280.  doi: 10.1215/S0012-7094-91-06311-8.

[27]

M. Walsh, Norm convergence of nilpotent ergodic averages, Annals of Math., 175 (2012), 1667-1688.  doi: 10.4007/annals.2012.175.3.15.

[28]

T. Ziegler, Universal characteristic factors and Furstenberg averages, J. Amer. Math. Soc., 20 (2007), 53-97.  doi: 10.1090/S0894-0347-06-00532-7.

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