July  2022, 42(7): 3379-3413. doi: 10.3934/dcds.2022019

Polynomial ergodic averages for certain countable ring actions

1. 

Beijing Institute of Mathematical Sciences and Applications, Beijing, China

2. 

Department of Mathematics, Nicolaus Copernicus University, Toruń, Poland

Received  October 2021 Revised  January 2022 Published  July 2022 Early access  March 2022

A recent result of Frantzikinakis in [17] establishes sufficient conditions for joint ergodicity in the setting of
$ \mathbb{Z} $
-actions. We generalize this result for actions of second-countable locally compact abelian groups. We obtain two applications of this result. First, we show that, given an ergodic action
$ (T_n)_{n \in F} $
of a countable field
$ F $
with characteristic zero on a probability space
$ (X,\mathcal{B},\mu) $
and a family
$ \{p_1,\dots,p_k\} $
of independent polynomials, we have
$ \lim\limits_{N \to \infty} \frac{1}{|\Phi_N|}\sum\limits_{n \in \Phi_N} T_{p_1(n)}f_1\cdots T_{p_k(n)}f_k\ = \ \prod\limits_{j = 1}^k \int_X f_i \ d\mu, $
where
$ f_i \in L^{\infty}(\mu) $
,
$ (\Phi_N) $
is a Følner sequence of
$ (F,+) $
, and the convergence takes place in
$ L^2(\mu) $
. This yields corollaries in combinatorics and topological dynamics. Second, we prove that a similar result holds for totally ergodic actions of suitable rings.
Citation: Andrew Best, Andreu Ferré Moragues. Polynomial ergodic averages for certain countable ring actions. Discrete and Continuous Dynamical Systems, 2022, 42 (7) : 3379-3413. doi: 10.3934/dcds.2022019
References:
[1]

E. Ackelsberg and V. Bergelson, Polynomial multiple recurrence and large intersections in rings of integers, preprint, arXiv: 2107.07626.

[2]

E. Ackelsberg, V. Bergelson and A. Best, Multiple recurrence and large intersections for abelian group actions, Discrete Anal., 2021, Paper No. 18, 91 pp. doi: 10.19086/da.

[3]

M. BeiglböckV. Bergelson and A. Fish, Sumset phenomenon in countable amenable groups, Adv. Math., 223 (2010), 416-432.  doi: 10.1016/j.aim.2009.08.009.

[4]

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

[5]

V. Bergelson, Ergodic Ramsey theory—an update, in Ergodic Theory of Zd Actions (Warwick, 1993–1994), 1–61, London Math. Soc. Lecture Note Ser., 228, Cambridge Univ. Press, Cambridge, 1996. doi: 10.1017/CBO9780511662812.002.

[6]

V. Bergelson, Ergodic theory and Diophantine problems, in Topics in Symbolic Dynamics and Applications (Temuco, 1997), 167–205, London Math. Soc. Lecture Note Ser., 279, Cambridge Univ. Press, Cambridge, 2000.

[7]

V. Bergelson and A. Ferré Moragues, An ergodic correspondence principle, invariant means and applications, Israel J. Math., 245 (2021), 921-962.  doi: 10.1007/s11856-021-2233-y.

[8]

V. Bergelson and A. Leibman, Cubic averages and large intersections, in Recent Trends in Ergodic Theory and Dynamical Systems, 5–19, Contemp. Math., 631, Amer. Math. Soc., Providence, RI, 2015. doi: 10.1090/conm/631/12592.

[9]

V. BergelsonA. Leibman and R. McCutcheon, Polynomial Szemerédi theorems for countable modules over integral domains and finite fields, J. Anal. Math., 95 (2005), 243-296.  doi: 10.1007/BF02791504.

[10]

V. Bergelson and J. Moreira, Van der Corput's difference theorem: Some modern developments, Indag. Math. (N.S.), 27 (2016), 437-479.  doi: 10.1016/j.indag.2015.10.014.

[11]

V. BergelsonT. Tao and T. Ziegler, Multiple recurrence and convergence results associated to $\mathbb{F}_p^\omega$-actions, J. Anal. Math., 127 (2015), 329-378.  doi: 10.1007/s11854-015-0033-1.

[12]

P. Durcik, R. Greenfeld, A. Iseli, A. Jamneshan and J. Madrid, An uncountable ergodic Roth theorem and applications, arXiv: 2101.00685.

[13]

T. Eisner, B. Farkas, M. Haase and R. Nagel, Operator Theoretic Aspects of Ergodic Theory, Graduate Texts in Mathematics, 272, Springer, Cham, 2015. doi: 10.1007/978-3-319-16898-2.

[14]

G. B. Folland, A Course in Abstract Harmonic Analysis, Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1995.

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

[17]

N. Frantzikinakis, Joint ergodicity of sequences, arXiv: 2102.09967.

[18]

N. Frantzikinakis and B. Kra, Polynomial averages converge to the product of integrals, Israel J. Math., 148 (2005), 267-276.  doi: 10.1007/BF02775439.

[19]

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.

[20]

W. HuangS. Shao and X. Ye, Topological correspondence of multiple ergodic averages of nilpotent group actions, J. Anal. Math., 138 (2019), 687-715.  doi: 10.1007/s11854-019-0036-4.

[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]

P. G. Larick, Results in Polynomial Recurrence for Actions of Fields, Ph.D thesis, The Ohio State University, 1998.

[23]

A. Leibman, Convergence of multiple ergodic averages along polynomials of several variables, Israel J. Math., 146 (2005), 303-315.  doi: 10.1007/BF02773538.

[24]

S. Peluse, On the polynomial Szemerédi theorem in finite fields, Duke Math. J., 168 (2019), 749-774.  doi: 10.1215/00127094-2018-0051.

[25]

S. Peluse and S. Prendiville, Quantitative bounds in the non-linear Roth theorem, arXiv: 1903.02592.

[26]

O. Shalom, Multiple ergodic averages in abelian groups and Khintchine type recurrence, Trans. Amer. Math. Soc., Published electronically: December 20, 2021. doi: 10.1090/tran/8558.

[27]

P. Zorin-Kranich, Norm convergence of multiple ergodic averages on amenable groups, J. Anal. Math., 130 (2016), 219-241.  doi: 10.1007/s11854-016-0035-7.

show all references

References:
[1]

E. Ackelsberg and V. Bergelson, Polynomial multiple recurrence and large intersections in rings of integers, preprint, arXiv: 2107.07626.

[2]

E. Ackelsberg, V. Bergelson and A. Best, Multiple recurrence and large intersections for abelian group actions, Discrete Anal., 2021, Paper No. 18, 91 pp. doi: 10.19086/da.

[3]

M. BeiglböckV. Bergelson and A. Fish, Sumset phenomenon in countable amenable groups, Adv. Math., 223 (2010), 416-432.  doi: 10.1016/j.aim.2009.08.009.

[4]

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

[5]

V. Bergelson, Ergodic Ramsey theory—an update, in Ergodic Theory of Zd Actions (Warwick, 1993–1994), 1–61, London Math. Soc. Lecture Note Ser., 228, Cambridge Univ. Press, Cambridge, 1996. doi: 10.1017/CBO9780511662812.002.

[6]

V. Bergelson, Ergodic theory and Diophantine problems, in Topics in Symbolic Dynamics and Applications (Temuco, 1997), 167–205, London Math. Soc. Lecture Note Ser., 279, Cambridge Univ. Press, Cambridge, 2000.

[7]

V. Bergelson and A. Ferré Moragues, An ergodic correspondence principle, invariant means and applications, Israel J. Math., 245 (2021), 921-962.  doi: 10.1007/s11856-021-2233-y.

[8]

V. Bergelson and A. Leibman, Cubic averages and large intersections, in Recent Trends in Ergodic Theory and Dynamical Systems, 5–19, Contemp. Math., 631, Amer. Math. Soc., Providence, RI, 2015. doi: 10.1090/conm/631/12592.

[9]

V. BergelsonA. Leibman and R. McCutcheon, Polynomial Szemerédi theorems for countable modules over integral domains and finite fields, J. Anal. Math., 95 (2005), 243-296.  doi: 10.1007/BF02791504.

[10]

V. Bergelson and J. Moreira, Van der Corput's difference theorem: Some modern developments, Indag. Math. (N.S.), 27 (2016), 437-479.  doi: 10.1016/j.indag.2015.10.014.

[11]

V. BergelsonT. Tao and T. Ziegler, Multiple recurrence and convergence results associated to $\mathbb{F}_p^\omega$-actions, J. Anal. Math., 127 (2015), 329-378.  doi: 10.1007/s11854-015-0033-1.

[12]

P. Durcik, R. Greenfeld, A. Iseli, A. Jamneshan and J. Madrid, An uncountable ergodic Roth theorem and applications, arXiv: 2101.00685.

[13]

T. Eisner, B. Farkas, M. Haase and R. Nagel, Operator Theoretic Aspects of Ergodic Theory, Graduate Texts in Mathematics, 272, Springer, Cham, 2015. doi: 10.1007/978-3-319-16898-2.

[14]

G. B. Folland, A Course in Abstract Harmonic Analysis, Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1995.

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

[17]

N. Frantzikinakis, Joint ergodicity of sequences, arXiv: 2102.09967.

[18]

N. Frantzikinakis and B. Kra, Polynomial averages converge to the product of integrals, Israel J. Math., 148 (2005), 267-276.  doi: 10.1007/BF02775439.

[19]

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.

[20]

W. HuangS. Shao and X. Ye, Topological correspondence of multiple ergodic averages of nilpotent group actions, J. Anal. Math., 138 (2019), 687-715.  doi: 10.1007/s11854-019-0036-4.

[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]

P. G. Larick, Results in Polynomial Recurrence for Actions of Fields, Ph.D thesis, The Ohio State University, 1998.

[23]

A. Leibman, Convergence of multiple ergodic averages along polynomials of several variables, Israel J. Math., 146 (2005), 303-315.  doi: 10.1007/BF02773538.

[24]

S. Peluse, On the polynomial Szemerédi theorem in finite fields, Duke Math. J., 168 (2019), 749-774.  doi: 10.1215/00127094-2018-0051.

[25]

S. Peluse and S. Prendiville, Quantitative bounds in the non-linear Roth theorem, arXiv: 1903.02592.

[26]

O. Shalom, Multiple ergodic averages in abelian groups and Khintchine type recurrence, Trans. Amer. Math. Soc., Published electronically: December 20, 2021. doi: 10.1090/tran/8558.

[27]

P. Zorin-Kranich, Norm convergence of multiple ergodic averages on amenable groups, J. Anal. Math., 130 (2016), 219-241.  doi: 10.1007/s11854-016-0035-7.

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