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  2021, 17: 1-32. doi: 10.3934/jmd.2021001

On mixing and sparse ergodic theorems

Asaf Katz 1,2,

1. 

Einstein Institute of Mathematics, The Hebrew University of Jerusalem, Jerusalem, 91904, Israel
2. 

Department of Mathematics, University of Michigan, Ann Arbor, MI 48109, USA

Received  February 15, 2018 Revised  October 14, 2020

We consider Bourgain's ergodic theorem regarding arithmetic averages in the cases where quantitative mixing is present in the dynamical system. Focusing on the case of the horocyclic flow, those estimates allow us to bound from above the Hausdorff dimension of the exceptional set, providing evidence towards conjectures by Margulis, Shah, and Sarnak regarding equidistribution of arithmetic averages in homogeneous spaces. We also prove the existence of a uniform upper bound for the Hausdorff dimension of the exceptional set which is independent of the spectral gap.

Keywords: Sparse ergodic theorem, sparse equidistribution, unipotent flows.
Mathematics Subject Classification: Primary: 37A05; Secondary: 37A17, 37A25.
Citation: Asaf Katz. On mixing and sparse ergodic theorems. Journal of Modern Dynamics, 2021, 17: 1-32. doi: 10.3934/jmd.2021001
References:
[1]

J. Bourgain, An approach to pointwise ergodic theorems, in Geometric Aspects of Functional Analysis (1986/87), Lecture Notes in Math., 1317, Springer, Berlin, 1988,204–223. doi: 10.1007/BFb0081742.  Google Scholar

[2]

J. Bourgain, Pointwise ergodic theorems for arithmetic sets, with an appendix by the author, H. Furstenberg, Y. Katznelson and D. S. Ornstein, Inst. Hautes Études Sci. Publ. Math., 69 (1989), 5-45.   Google Scholar

[3]

J. Bourgain, On the Vinogradov integral, Proc. Steklov Inst. Math., 296 (2017), no. 1, 30–40. doi: 10.1134/S0371968517010034.  Google Scholar

[4]

D. Bump, Automorphic Forms and Representations, Cambridge Studies in Advanced Mathematics, 55, Cambridge University Press, Cambridge, 1997. doi: 10.1017/CBO9780511609572.  Google Scholar

[5]

M. Burger, Horocycle flow on geometrically finite surfaces, Duke Math. J., 61 (1990), 779-803.  doi: 10.1215/S0012-7094-90-06129-0.  Google Scholar

[6]

S. G. Dani and G. A. Margulis, Limit distributions of orbits of unipotent flows and values of quadratic forms, in I. M. Gel'fand Seminar, Adv. Soviet Math., 16, Amer. Math. Soc., Providence, RI, 1993, 91–137.  Google Scholar

[7]

S. G. Dani and J. Smillie, Uniform distribution of horocycle orbits for Fuchsian groups, Duke Math. J., 51 (1984), 185-194.  doi: 10.1215/S0012-7094-84-05110-X.  Google Scholar

[8]

L. Flaminio and G. Forni, Invariant distributions and time averages for horocycle flows, Duke Math. J., 119 (2003), 465-526.  doi: 10.1215/S0012-7094-03-11932-8.  Google Scholar

[9]

L. Flaminio, G. Forni and J. Tanis, Effective equidistribution of twisted horocycle flows and horocycle maps, preprint, arXiv: 1507.05147. doi: 10.1007/s00039-016-0385-4.  Google Scholar

[10]

I. M. Gel'fand, M. I. Graev, and I. I. Pyatetskii-Shapiro, Representation Theory and Automorphic Functions, Generalized Functions, 6, Academic Press, Inc., Boston, MA, 1990.  Google Scholar

[11]

B. Green and T. Tao, The quantitative behaviour of polynomial orbits on nilmanifolds, Ann. of Math. (2), 175 (2012), 465-540.  doi: 10.4007/annals.2012.175.2.2.  Google Scholar

[12]

H. Halberstam and H.-E. Richert, Sieve Methods, London Mathematical Society Monographs, 4, Academic Press, London-New York, 1974.  Google Scholar

[13]

Ha rish-Chandra, Discrete series for semisimple lie groups. Ⅱ: Explicit determination of the characters, Acta Math., 116 (1966), 1-111.  doi: 10.1007/BF02392813.  Google Scholar

[14]

L.-K. Hua, On Waring's problem, The Quarterly Journal of Mathematics, os-9 (1938), 199-202.  doi: 10.1093/qmath/os-9.1.199.  Google Scholar

[15]

H. Iwaniec and E. Kowalski, Analytic Number Theory, American Mathematical Society Colloquium Publications, 53, American Mathematical Society, Providence, RI, 2004. doi: 10.1090/coll/053.  Google Scholar

[16]

Y. Katznelson, An Introduction to Harmonic Analysis, 3rd edition, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 2004. doi: 10.1017/CBO9781139165372.  Google Scholar

[17]

H. Kim and P. Sarnak, Refined estimates towards the Ramanujan and Selberg conjectures, J. Amer. Math. Soc., 16 (2003), 175-181.   Google Scholar

[18]

A. W. Knapp, Representation Theory of Semisimple Groups: An Overview Based on Examples, Princeton Landmarks in Mathematics, Princeton University Press, Princeton, NJ, 2001.  Google Scholar

[19]

A. W. Knapp, Lie Groups Beyond an Introduction, 2$^nd$ edition, Progress in Mathematics, 140, Birkhäuser Boston, Inc., Boston, MA, 2002.  Google Scholar

[20]

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.  Google Scholar

[21]

G. Margulis, Problems and conjectures in rigidity theory, in Mathematics: Frontiers and Perspectives, Amer. Math. Soc., Providence, RI, 2000,161–174.  Google Scholar

[22]

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

[23]

J. M. Rosenblatt and M. Wierdl, Pointwise ergodic theorems via harmonic analysis, in Ergodic Theory and its Connections with Harmonic Analysis (Alexandria, 1993), London Math. Soc. Lecture Note Ser., 205, Cambridge Univ. Press, Cambridge, 1995, 3–151. doi: 10.1017/CBO9780511574818.002.  Google Scholar

[24]

P. Sarnak, Möbius randomness and dynamics, Not. S. Afr. Math. Soc., 43 (2012), 89-97.   Google Scholar

[25]

P. Sarnak and A. Ubis, The horocycle flow at prime times, J. Math. Pures Appl. (9), 103 (2015), 575-618.  doi: 10.1016/j.matpur.2014.07.004.  Google Scholar

[26]

N. A. Shah, Limit distributions of polynomial trajectories on homogeneous spaces, Duke Math. J., 75 (1994), 711-732.  doi: 10.1215/S0012-7094-94-07521-2.  Google Scholar

[27]

A. Strömbergsson, On the deviation of ergodic averages for horocycle flows, J. Mod. Dyn., 7 (2013), 291-328.  doi: 10.3934/jmd.2013.7.291.  Google Scholar

[28]

J. Tanis and P. Vishe, Uniform bounds for period integrals and sparse equidistribution, Int. Math. Res. Not. IMRN, 24 (2015), 13728-13756.  doi: 10.1093/imrn/rnv115.  Google Scholar

[29]

A. Venkatesh, Sparse equidistribution problems, period bounds and subconvexity, Ann. of Math. (2), 172 (2010), 989-1094.  doi: 10.4007/annals.2010.172.989.  Google Scholar

[30]

G. N. Watson, A Treatise on the Theory of Bessel Functions, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1995.  Google Scholar

[31]

C. Zheng, Sparse equidistribution of unipotent orbits in finite-volume quotients of ${\text PSL}(2, \mathbb{R})$, J. Mod. Dyn., 10 (2016), 1-21.  doi: 10.3934/jmd.2016.10.1.  Google Scholar

show all references

References:
[1]

J. Bourgain, An approach to pointwise ergodic theorems, in Geometric Aspects of Functional Analysis (1986/87), Lecture Notes in Math., 1317, Springer, Berlin, 1988,204–223. doi: 10.1007/BFb0081742.  Google Scholar

[2]

J. Bourgain, Pointwise ergodic theorems for arithmetic sets, with an appendix by the author, H. Furstenberg, Y. Katznelson and D. S. Ornstein, Inst. Hautes Études Sci. Publ. Math., 69 (1989), 5-45.   Google Scholar

[3]

J. Bourgain, On the Vinogradov integral, Proc. Steklov Inst. Math., 296 (2017), no. 1, 30–40. doi: 10.1134/S0371968517010034.  Google Scholar

[4]

D. Bump, Automorphic Forms and Representations, Cambridge Studies in Advanced Mathematics, 55, Cambridge University Press, Cambridge, 1997. doi: 10.1017/CBO9780511609572.  Google Scholar

[5]

M. Burger, Horocycle flow on geometrically finite surfaces, Duke Math. J., 61 (1990), 779-803.  doi: 10.1215/S0012-7094-90-06129-0.  Google Scholar

[6]

S. G. Dani and G. A. Margulis, Limit distributions of orbits of unipotent flows and values of quadratic forms, in I. M. Gel'fand Seminar, Adv. Soviet Math., 16, Amer. Math. Soc., Providence, RI, 1993, 91–137.  Google Scholar

[7]

S. G. Dani and J. Smillie, Uniform distribution of horocycle orbits for Fuchsian groups, Duke Math. J., 51 (1984), 185-194.  doi: 10.1215/S0012-7094-84-05110-X.  Google Scholar

[8]

L. Flaminio and G. Forni, Invariant distributions and time averages for horocycle flows, Duke Math. J., 119 (2003), 465-526.  doi: 10.1215/S0012-7094-03-11932-8.  Google Scholar

[9]

L. Flaminio, G. Forni and J. Tanis, Effective equidistribution of twisted horocycle flows and horocycle maps, preprint, arXiv: 1507.05147. doi: 10.1007/s00039-016-0385-4.  Google Scholar

[10]

I. M. Gel'fand, M. I. Graev, and I. I. Pyatetskii-Shapiro, Representation Theory and Automorphic Functions, Generalized Functions, 6, Academic Press, Inc., Boston, MA, 1990.  Google Scholar

[11]

B. Green and T. Tao, The quantitative behaviour of polynomial orbits on nilmanifolds, Ann. of Math. (2), 175 (2012), 465-540.  doi: 10.4007/annals.2012.175.2.2.  Google Scholar

[12]

H. Halberstam and H.-E. Richert, Sieve Methods, London Mathematical Society Monographs, 4, Academic Press, London-New York, 1974.  Google Scholar

[13]

Ha rish-Chandra, Discrete series for semisimple lie groups. Ⅱ: Explicit determination of the characters, Acta Math., 116 (1966), 1-111.  doi: 10.1007/BF02392813.  Google Scholar

[14]

L.-K. Hua, On Waring's problem, The Quarterly Journal of Mathematics, os-9 (1938), 199-202.  doi: 10.1093/qmath/os-9.1.199.  Google Scholar

[15]

H. Iwaniec and E. Kowalski, Analytic Number Theory, American Mathematical Society Colloquium Publications, 53, American Mathematical Society, Providence, RI, 2004. doi: 10.1090/coll/053.  Google Scholar

[16]

Y. Katznelson, An Introduction to Harmonic Analysis, 3rd edition, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 2004. doi: 10.1017/CBO9781139165372.  Google Scholar

[17]

H. Kim and P. Sarnak, Refined estimates towards the Ramanujan and Selberg conjectures, J. Amer. Math. Soc., 16 (2003), 175-181.   Google Scholar

[18]

A. W. Knapp, Representation Theory of Semisimple Groups: An Overview Based on Examples, Princeton Landmarks in Mathematics, Princeton University Press, Princeton, NJ, 2001.  Google Scholar

[19]

A. W. Knapp, Lie Groups Beyond an Introduction, 2$^nd$ edition, Progress in Mathematics, 140, Birkhäuser Boston, Inc., Boston, MA, 2002.  Google Scholar

[20]

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.  Google Scholar

[21]

G. Margulis, Problems and conjectures in rigidity theory, in Mathematics: Frontiers and Perspectives, Amer. Math. Soc., Providence, RI, 2000,161–174.  Google Scholar

[22]

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

[23]

J. M. Rosenblatt and M. Wierdl, Pointwise ergodic theorems via harmonic analysis, in Ergodic Theory and its Connections with Harmonic Analysis (Alexandria, 1993), London Math. Soc. Lecture Note Ser., 205, Cambridge Univ. Press, Cambridge, 1995, 3–151. doi: 10.1017/CBO9780511574818.002.  Google Scholar

[24]

P. Sarnak, Möbius randomness and dynamics, Not. S. Afr. Math. Soc., 43 (2012), 89-97.   Google Scholar

[25]

P. Sarnak and A. Ubis, The horocycle flow at prime times, J. Math. Pures Appl. (9), 103 (2015), 575-618.  doi: 10.1016/j.matpur.2014.07.004.  Google Scholar

[26]

N. A. Shah, Limit distributions of polynomial trajectories on homogeneous spaces, Duke Math. J., 75 (1994), 711-732.  doi: 10.1215/S0012-7094-94-07521-2.  Google Scholar

[27]

A. Strömbergsson, On the deviation of ergodic averages for horocycle flows, J. Mod. Dyn., 7 (2013), 291-328.  doi: 10.3934/jmd.2013.7.291.  Google Scholar

[28]

J. Tanis and P. Vishe, Uniform bounds for period integrals and sparse equidistribution, Int. Math. Res. Not. IMRN, 24 (2015), 13728-13756.  doi: 10.1093/imrn/rnv115.  Google Scholar

[29]

A. Venkatesh, Sparse equidistribution problems, period bounds and subconvexity, Ann. of Math. (2), 172 (2010), 989-1094.  doi: 10.4007/annals.2010.172.989.  Google Scholar

[30]

G. N. Watson, A Treatise on the Theory of Bessel Functions, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1995.  Google Scholar

[31]

C. Zheng, Sparse equidistribution of unipotent orbits in finite-volume quotients of ${\text PSL}(2, \mathbb{R})$, J. Mod. Dyn., 10 (2016), 1-21.  doi: 10.3934/jmd.2016.10.1.  Google Scholar

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