March  2021, 41(3): 1177-1205. doi: 10.3934/dcds.2020314

Multiple ergodic averages for tempered functions

The Ohio State University, Department of Mathematics, Columbus, Ohio, USA

Received  May 2020 Revised  July 2020 Published  March 2021 Early access  August 2020

Following Frantzikinakis' approach on averages for Hardy field functions of different growth, we add to the topic by studying the corresponding averages for tempered functions, a class which also contains functions that oscillate and is in general more restrictive to deal with. Our main result is the existence and the explicit expression of the $ L^2 $-norm limit of the aforementioned averages, which turns out, as in the Hardy field case, to be the "expected" one. The main ingredients are the use of, the now classical, PET induction (introduced by Bergelson), covering a more general case, namely a "nice" class of tempered functions (developed by Chu-Frantzikinakis-Host for polynomials and Frantzikinakis for Hardy field functions) and some equidistribution results on nilmanifolds (analogous to the ones of Frantzikinakis' for the Hardy field case).

Citation: Andreas Koutsogiannis. Multiple ergodic averages for tempered functions. Discrete and Continuous Dynamical Systems, 2021, 41 (3) : 1177-1205. doi: 10.3934/dcds.2020314
References:
[1]

T. Austin, Pleasant extensions retaining algebraic structure, II, J. Anal. Math., 126 (2015), 1-111.  doi: 10.1007/s11854-015-0013-5.

[2]

V. Bergelson, Ergodic Ramsey theory, in Logic and Combinatorics, Contemp. Math., 65, Amer. Math. Soc., Providence, RI, 1987, 63–87.

[3]

V. Bergelson, Ergodic Ramsey theory – An update, in Ergodic Theory of $\mathbb{Z}^d$-Actions, London Math. Soc. Lecture Note Ser., 228, Cambridge Univ. Press, Cambridge, 1996, 1–61. doi: 10.1017/CBO9780511662812.002.

[4]

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

[5]

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

[6]

V. BergelsonB. Host and B. Kra, Multiple recurrence and nilsequences, Invent. Math., 160 (2005), 261-303.  doi: 10.1007/s00222-004-0428-6.

[7]

V. BergelsonG. Kolesnik and Y. Son, Uniform distribution of subpolynomial functions along primes and applications, J. Anal. Math., 137 (2019), 135-187.  doi: 10.1007/s11854-018-0068-1.

[8]

V. Bergelson and A. Leibman, Distribution of values of bounded generalized polynomials, Acta Math., 198 (2007), 155-230.  doi: 10.1007/s11511-007-0015-y.

[9]

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.

[10]

Q. Chu, N. Frantzikinakis and B. Host, Ergodic averages of commuting transformations with distinct degree polynomial iterates, Proc. Lond. Math. Soc. (3), 102 (2011), 801–842. doi: 10.1112/plms/pdq037.

[11]

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.

[12]

S. Donoso, A. Koutsogiannis and W. Sun, Seminorms for multiple averages along polynomials and applications to joint ergodicity, To appear in J. Anal. Math.

[13]

N. Frantzikinakis, Multiple correlation sequences and nilsequences, Invent. Math., 202 (2015), 875-892.  doi: 10.1007/s00222-015-0579-7.

[14]

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.

[15]

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.

[16]

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

[17]

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

[18]

H. FurstenbergY. Katznelson and D. Ornstein, The ergodic theoretical proof of Szemerédi's theorem, Bull. Amer. Math. Soc. (N. S.), 7 (1982), 527-552.  doi: 10.1090/S0273-0979-1982-15052-2.

[19]

B. Host and B. Kra, Nonconventional ergodic averages and nilmanifolds, Ann. of Math. (2), 161 (2005), 397–488. doi: 10.4007/annals.2005.161.397.

[20]

B. Host and B. Kra, Uniformity seminorms on $l^{\infty}$ and applications, J. Anal. Math., 108 (2009), 219-276.  doi: 10.1007/s11854-009-0024-1.

[21]

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.

[22]

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

[23]

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

[24]

L. Kuipers and H. Niederreiter, Uniform Distribution of Sequences, Pure and Applied Mathematics, Wiley-Interscience, New York-London-Sydney, 1974.

[25]

A. Leibman, Multiple polynomial correlation sequences and nilsequences, Ergodic Theory Dynam. Systems, 30 (2010), 841-854.  doi: 10.1017/S0143385709000303.

[26]

A. Leibman, Nilsequences, null-sequences, and multiple correlation sequences, Ergodic Theory Dynam. Systems, 35 (2015), 176-191.  doi: 10.1017/etds.2013.36.

[27]

M. N. Walsh, Norm convergence of nilpotent ergodic averages, Ann. of Math. (2), 175 (2012), 1667–1688. doi: 10.4007/annals.2012.175.3.15.

[28]

H. Weyl, Über die Gleichverteilung von Zahlen mod. Eins., Math. Ann., 77 (1916), 313-352.  doi: 10.1007/BF01475864.

show all references

References:
[1]

T. Austin, Pleasant extensions retaining algebraic structure, II, J. Anal. Math., 126 (2015), 1-111.  doi: 10.1007/s11854-015-0013-5.

[2]

V. Bergelson, Ergodic Ramsey theory, in Logic and Combinatorics, Contemp. Math., 65, Amer. Math. Soc., Providence, RI, 1987, 63–87.

[3]

V. Bergelson, Ergodic Ramsey theory – An update, in Ergodic Theory of $\mathbb{Z}^d$-Actions, London Math. Soc. Lecture Note Ser., 228, Cambridge Univ. Press, Cambridge, 1996, 1–61. doi: 10.1017/CBO9780511662812.002.

[4]

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

[5]

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

[6]

V. BergelsonB. Host and B. Kra, Multiple recurrence and nilsequences, Invent. Math., 160 (2005), 261-303.  doi: 10.1007/s00222-004-0428-6.

[7]

V. BergelsonG. Kolesnik and Y. Son, Uniform distribution of subpolynomial functions along primes and applications, J. Anal. Math., 137 (2019), 135-187.  doi: 10.1007/s11854-018-0068-1.

[8]

V. Bergelson and A. Leibman, Distribution of values of bounded generalized polynomials, Acta Math., 198 (2007), 155-230.  doi: 10.1007/s11511-007-0015-y.

[9]

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.

[10]

Q. Chu, N. Frantzikinakis and B. Host, Ergodic averages of commuting transformations with distinct degree polynomial iterates, Proc. Lond. Math. Soc. (3), 102 (2011), 801–842. doi: 10.1112/plms/pdq037.

[11]

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.

[12]

S. Donoso, A. Koutsogiannis and W. Sun, Seminorms for multiple averages along polynomials and applications to joint ergodicity, To appear in J. Anal. Math.

[13]

N. Frantzikinakis, Multiple correlation sequences and nilsequences, Invent. Math., 202 (2015), 875-892.  doi: 10.1007/s00222-015-0579-7.

[14]

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.

[15]

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.

[16]

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

[17]

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

[18]

H. FurstenbergY. Katznelson and D. Ornstein, The ergodic theoretical proof of Szemerédi's theorem, Bull. Amer. Math. Soc. (N. S.), 7 (1982), 527-552.  doi: 10.1090/S0273-0979-1982-15052-2.

[19]

B. Host and B. Kra, Nonconventional ergodic averages and nilmanifolds, Ann. of Math. (2), 161 (2005), 397–488. doi: 10.4007/annals.2005.161.397.

[20]

B. Host and B. Kra, Uniformity seminorms on $l^{\infty}$ and applications, J. Anal. Math., 108 (2009), 219-276.  doi: 10.1007/s11854-009-0024-1.

[21]

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.

[22]

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

[23]

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

[24]

L. Kuipers and H. Niederreiter, Uniform Distribution of Sequences, Pure and Applied Mathematics, Wiley-Interscience, New York-London-Sydney, 1974.

[25]

A. Leibman, Multiple polynomial correlation sequences and nilsequences, Ergodic Theory Dynam. Systems, 30 (2010), 841-854.  doi: 10.1017/S0143385709000303.

[26]

A. Leibman, Nilsequences, null-sequences, and multiple correlation sequences, Ergodic Theory Dynam. Systems, 35 (2015), 176-191.  doi: 10.1017/etds.2013.36.

[27]

M. N. Walsh, Norm convergence of nilpotent ergodic averages, Ann. of Math. (2), 175 (2012), 1667–1688. doi: 10.4007/annals.2012.175.3.15.

[28]

H. Weyl, Über die Gleichverteilung von Zahlen mod. Eins., Math. Ann., 77 (1916), 313-352.  doi: 10.1007/BF01475864.

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