Geodesic
On a multi-parameter variant of the Bellow–Furstenberg problem
Forum of Mathematics, Pi, Tome 11 (2023)

Voir la notice de l'article provenant de la source Cambridge University Press

We prove convergence in norm and pointwise almost everywhere on $L^p$, $p\in (1,\infty )$, for certain multi-parameter polynomial ergodic averages by establishing the corresponding multi-parameter maximal and oscillation inequalities. Our result, in particular, gives an affirmative answer to a multi-parameter variant of the Bellow–Furstenberg problem. This paper is also the first systematic treatment of multi-parameter oscillation semi-norms which allows an efficient handling of multi-parameter pointwise convergence problems with arithmetic features. The methods of proof of our main result develop estimates for multi-parameter exponential sums, as well as introduce new ideas from the so-called multi-parameter circle method in the context of the geometry of backwards Newton diagrams that are dictated by the shape of the polynomials defining our ergodic averages. 
@article{10_1017_fmp_2023_21,
     author = {Jean Bourgain and Mariusz Mirek and Elias M. Stein and James Wright},
     title = {On a multi-parameter variant of the {Bellow{\textendash}Furstenberg} problem},
     journal = {Forum of Mathematics, Pi},
     publisher = {mathdoc},
     volume = {11},
     year = {2023},
     doi = {10.1017/fmp.2023.21},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.1017/fmp.2023.21/}
}
TY  - JOUR
AU  - Jean Bourgain
AU  - Mariusz Mirek
AU  - Elias M. Stein
AU  - James Wright
TI  - On a multi-parameter variant of the Bellow–Furstenberg problem
JO  - Forum of Mathematics, Pi
PY  - 2023
VL  - 11
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/articles/10.1017/fmp.2023.21/
DO  - 10.1017/fmp.2023.21
LA  - en
ID  - 10_1017_fmp_2023_21
ER  - 
%0 Journal Article
%A Jean Bourgain
%A Mariusz Mirek
%A Elias M. Stein
%A James Wright
%T On a multi-parameter variant of the Bellow–Furstenberg problem
%J Forum of Mathematics, Pi
%D 2023
%V 11
%I mathdoc
%U http://geodesic.mathdoc.fr/articles/10.1017/fmp.2023.21/
%R 10.1017/fmp.2023.21
%G en
%F 10_1017_fmp_2023_21
Jean Bourgain; Mariusz Mirek; Elias M. Stein; James Wright. On a multi-parameter variant of the Bellow–Furstenberg problem. Forum of Mathematics, Pi, Tome 11 (2023). doi: 10.1017/fmp.2023.21

[1] Arkhipov, G. I., Chubarikov, V. N. and Karatsuba, A. A., Trigonometric Sums in Number Theory and Analysis (De Gruyter Expositions in Mathematics) (Walter De Gruyter, 2004).Google Scholar

[2] Arkhipov, G. I., Chubarikov, V. N. and Karatsuba, A. A., ‘Distribution of fractional parts of polynomials of several variable’, Mat. Zametki 25(1) (1979), 3–14 Google Scholar

[3] Austin, T., ‘A proof of Walsh’s convergence theorem using couplings’, Int. Math. Res. Not. IMRN 15 (2015), 6661–6674.Google Scholar | DOI

[4] Austin, T., ‘On the norm convergence of non-conventional ergodic averages’, Ergodic Theory Dynam. Systems 30 (2010), 321–338.Google Scholar | DOI

[5] Bellow, A., Measure Theory: Proceedings of the Conference Held at Oberwolfach,June 21–27, 1981 (Lecture Notes in Mathematics) vol. 541 (Springer-Verlag, Berlin, 1982). Section: Two problems submitted by A. Bellow, 429–431.Google Scholar | DOI

[6] Bergelson, V., ‘Weakly mixing PET’, Ergodic Theory Dynam. Systems 7(3) (1987), 337–349.Google Scholar | DOI

[7] Bergelson, V., ‘Ergodic Ramsey Theory – An Update’, in Pollicott, M. and Schmidt, K. (eds.) Ergodic Theory of -actions (London Math. Soc. Lecture Note Series) vol. 228 (1996), 1–61.Google Scholar

[8] Bergelson, V., ‘Combinatorial and diophantine applications of ergodic theory’, in Hasselblatt, B. and Katok, A. (eds.) Handbook of Dynamical Systems vol. 1B (Elsevier, 2006), 745–841.Google Scholar

[9] Bergelson, V. and Leibman, A., ‘Polynomial extensions of van derWaerden’s and Szemerédi’s theorems’, J. Amer. Math. Soc. 9 (1996), 725–753.Google Scholar | DOI

[10] Bergelson, V. and Leibman, A., ‘A nilpotent Roth theorem’, Invent. Math. 147 (2002), 429–470.Google Scholar | DOI

[11] Birkhoff, G., ‘Proof of the ergodic theorem’, Proc. Natl. Acad. Sci. USA 17(12) (1931), 656–660.Google ScholarPubMed | DOI

[12] Bohl, P., ‘Über ein in der Theorie der säkularen Störungen vorkommendes Problem’, J. Reine Angew. Math. 135 (1909), 189–283.Google Scholar | DOI

[13] Bourgain, J., ‘On the maximal ergodic theorem for certain subsets of the integers’, Israel J. Math. 61 (1988), 39–72.Google Scholar | DOI

[14] Bourgain, J., ‘On the pointwise ergodic theorem on for arithmetic sets’, Israel J. Math. 61 (1988), 73–84.Google Scholar | DOI

[15] Bourgain, J., ‘Pointwise ergodic theorems for arithmetic sets’, Inst. Hautes Etudes Sci. Publ. Math. 69 (1989), 5–45. With an appendix by the author, H. Furstenberg, Y. Katznelson and D. S. Ornstein.Google Scholar | DOI

[16] Bourgain, J., ‘Double recurrence and almost sure convergence’, J. Reine Angew. Math. 404 (1990), 140–161.Google Scholar

[17] Bourgain, J., Demeter, C. and Guth, L., ‘Proof of the main conjecture in Vinogradov’s mean value theorem for degrees higher than three’, Ann. Math. 184(2) (2016), 633–682.Google Scholar | DOI

[18] Buczolich, Z. and Mauldin, R. D., ‘Divergent square averages’, Ann. Math. 171(3) (2010), 1479–1530.Google Scholar | DOI

[19] Calderón, A, ‘Ergodic theory and translation invariant operators’, Proc. Natl. Acad. Sci. USA 59 (1968), 349–353.Google ScholarPubMed | DOI

[20] Carbery, A., Christ, M. and Wright, J., ‘Multidimensional van der Corput and sublevel set estimates’, J. Amer. Math. Soc. 12(4) (1999), 3981–1015.Google Scholar | DOI

[21] Chu, Q., Frantzikinakis, N. and Host, B., ‘Ergodic averages of commuting transformations with distinct degree polynomial iterates’, Proc. London. Math. Soc. 102(5) (2011), 801–842.Google Scholar | DOI

[22] Duoandikoetxea, J. and Rubio De Francia, J. L., ‘Maximal and singular integral operators via Fourier transform estimates’, Invent. Math. 84 (1984), 541–562.Google Scholar | DOI

[23] Dunford, N., ‘An individual ergodic theorem for non-commutative transformations’, Acta Sci. Math. (Szeged) 14 (1951), 1–4.Google Scholar

[24] Furstenberg, H., Problems Session, Conference on Ergodic Theory and Applications University of New Hampshire, Durham, NH, June 1982.Google Scholar

[25] Frantzikinakis, N., ‘Some open problems on multiple ergodic averages’, Bull. Hellenic Math. Soc. 60 (2016), 41–90.Google Scholar

[26] Frantzikinakis, N. and Kra, B., ‘Polynomial averages converge to the product of integrals’, Israel J. Math. 148 (2005), 267–276.Google Scholar | DOI

[27] Furstenberg, H., ‘Ergodic behavior of diagonal measures and a theorem of Szemeredi on arithmetic progressions’, J. Anal. Math. 31 (1977), 204–256.Google Scholar | DOI

[28] Furstenberg, H., ‘Nonconventional ergodic averages’, in The Legacy of John von Neumann (Amer. Math. Soc., Providence, RI, 1990), 43–56.Google Scholar | DOI

[29] Furstenberg, H., Recurrence in Ergodic Theory and Combinatorial Number Theory (Princeton University Press, 1981).Google Scholar | DOI

[30] Furstenberg, H. and Weiss, B., ‘A mean ergodic theorem for ’, in Convergence in Ergodic Theory and Probability (de Gruyter, Berlin, 1996), 193–227.Google Scholar | DOI

[31] Host, B. and Kra, B., ‘Non-conventional ergodic averages and nilmanifolds’, Ann. Math. 161 (2005), 397–488.Google Scholar | DOI

[32] Host, B. and Kra, B., ‘Convergence of polynomial ergodic averages’, Israel J. Math. 149 (2005), 1–19.Google Scholar | DOI

[33] Ionescu, A. D., Magyar, A., Stein, E. M. and Wainger, S., ‘Discrete Radon transforms and applications to ergodic theory’, Acta Math. 198 (2007), 231–298.Google Scholar | DOI

[34] Ionescu, A. D. and Wainger, S., ‘ boundedness of discrete singular Radon transforms’, J. Amer. Math. Soc. 19(2) (2005), 357–383.Google Scholar | DOI

[35] Iwaniec, H. and Kowalski, E., Analytic Number Theory vol. 53 (Amer. Math. Soc. Colloquium Publications, Providence RI, 2004).Google Scholar

[36] Ionescu, A. D., Magyar, Á.S, Mirek, M. and Szarek, T. Z., ‘Polynomial averages and pointwise ergodic theorems on nilpotent groups’, Invent. Math. 231 (2023), 1023–1140.Google Scholar | DOI

[37] Jones, R. L., Rosenblatt, J. M. and Wierdl, M., ‘Oscillation inequalities for rectangles’, Proc. Amer. Math. Soc. 129(5) (2001), 1349–1358.Google Scholar | DOI

[38] Jones, R. L., Seeger, A. and Wright, J., ‘Strong variational and jump inequalities in harmonic analysis’, Trans. Amer. Math. Soc. 360(12) (2008), 6711–6742.Google Scholar | DOI

[39] Karatsuba, A. A., Basic Analytic Number Theory (Springer-Verlag, Berlin, 1993). Translated from the second (1983) Russian edition and with a preface by Nathanson, Melvyn B..Google Scholar | DOI

[40] Khintchin, A. Y., ‘Zur Birkhoff’s Lb’sung des Ergodensproblems’, Math. Ann. 107 (1933), 485–488.Google Scholar | DOI

[41] Krause, B., Mirek, M. and Tao, T., ‘Pointwise ergodic theorems for non-conventional bilinear polynomial averages’, Ann. Math. 195(3) (2022), 997–1109.Google Scholar | DOI

[42] Lavictoire, P., ‘Universally -bad arithmetic sequences’, J. Anal. Math. 113(1) (2011), 241–263.Google Scholar | DOI

[43] Leibman, A., ‘Convergence of multiple ergodic averages along polynomials of several variables’, Israel J. Math. 146 (2005), 303–315.Google Scholar | DOI

[44] Magyar, A., Stein, E. M. and Wainger, S., ‘Discrete analogues in harmonic analysis: spherical averages’, Ann. Math. 155 (2002), 189–208.Google Scholar | DOI

[45] Magyar, A., Stein, E. M. and Wainger, S., ‘Maximal operators associated to discrete subgroups of nilpotent Lie groups’, J. Anal. Math. 101(1) (2007), 257–312.Google Scholar | DOI

[46] Mirek, M., ‘-estimates for discrete Radon transform: square function estimates’, Anal. PDE 11(3) (2018), 583–608.Google Scholar | DOI

[47] Mirek, M., Slomian, W. and Szarek, T., ‘Some remarks on oscillation inequalities’, Ergodic Theory and Dynam. Systems, online version 29 (November 2022), 1–30.Google Scholar

[48] Mirek, M., Stein, E. M. and Trojan, B., ‘-estimates for discrete operators of Radon types I: maximal functions and vector-valued estimates’, J. Funct. Anal. 277(8) (2019), 2471–2521.Google Scholar | DOI

[49] Mirek, M., Stein, E. M. and Trojan, B., ‘-estimates for discrete operators of Radon type: variational estimates’, Invent. Math. 209(3) (2017), 665–748.Google Scholar | DOI

[50] Mirek, M., Stein, E. M. and Zorin-Kranich, P., ‘Jump inequalities via real interpolation’, Math. Ann. 376(1–2) (2020), 797–819.Google Scholar | DOI

[51] Mirek, M., Stein, E. M. and Zorin-Kranich, P.. ‘A bootstrapping approach to jump inequalities and their applications’, Analysis & PDE 13(2) (2020), 527–558.Google Scholar | DOI

[52] Mirek, M., Stein, E. M. and Zorin-Kranich, P., ‘Jump inequalities for translation-invariant operators of Radon type on ’, Advances in Mathematics 365(107065) (2020), 57.Google Scholar | DOI

[53] Mirek, M., Szarek, T. Z. and Wright, J., ‘Oscillation inequalities in ergodic theory and anaysis; one parameter and multi-parameter perspectives’, Rev. Mat. Iberoam. 38(7) (2022), 2249–2284.Google Scholar | DOI

[54] Mirek, M. and Trojan, B., ‘Discrete maximal functions in higher dimensions and applications to ergodic theory’, Amer. J. Math. 138(6) (2016), 1495–1532.Google Scholar | DOI

[55] Pierce, L. B., ‘On superorthogonality’, J. Geom. Anal. 31 (2021), 7096–7183.Google Scholar | DOI

[56] Sierpiński, W., ‘Sur la valeur asymptotique d’une certaine somm’, Bull. Intl. Acad. Polonaise des Sci. et des Lettres (Cracovie) series A (1910), 9–11.Google Scholar

[57] Stein, E. M., Harmonic Analysis (Princeton University Press, 1993).Google Scholar

[58] Stein, E. M. and Wainger, S.’ ‘Discrete analogues in harmonic analysis I: estimates for singular Radon transforms’, Amer. J. Math. 121 (1999), 1291–1336.Google Scholar | DOI

[59] Szemerédi, E., ‘On sets of integers containing no elements in arithmetic progression’, Acta Arith. 27 (1975), 199–245.Google Scholar | DOI

[60] Tao, T., ‘Equidistribution for multidimensional polynomial phases’, available at Terence Tao’s blog, 06 August 2015: terrytao.wordpress.com/2015/08/06/equidistribution-for-multidimensional-polynomial-phases/.Google Scholar

[61] Tao, T., ‘Norm convergence of multiple ergodic averages for commuting transformations’, Ergodic Theory Dynam. Systems 28 (2008), 657–688.Google Scholar | DOI

[62] Tao, T., ‘The Ionescu–Wainger multiplier theorem and the adeles’, Mathematika 63(3) (2021), 557–737.Google Scholar

[63] Vinogradov, I. M., The Method of Trigonometrical Sums in the Theory of Numbers (Interscience Publishers, New York, 1954).Google Scholar

[64] Von Neumann, J., ‘Proof of the quasi-ergodic hypothesis’, Proc. Natl. Acad. Sci. USA 18 (1932), 70–82.Google Scholar | DOI

[65] Walsh, M., ‘Norm convergence of nilpotent ergodic averages’, Ann. Math. 175(3) (2012), 1667–1688.Google Scholar | DOI

[66] Weyl, H., ‘Über die Gibbs’sce Erscheinung und verwandte Konvergenzphenomene’, Rendiconti del Circolo Matematico di Palermo 330 (1910), 377–407.Google Scholar | DOI

[67] Weyl, H., ‘Über die Gleichverteilung von Zahlen mod. Eins’, Math. Ann. 7 (1916), 313–352.Google Scholar | DOI

[68] Wooley, T. D., ‘Nested efficient congruencing and relatives of Vinogradov’s mean value theorem’, Proc. London Math. Soc . (3) 118(4) (2019), 942–1016.Google Scholar | DOI

[69] Wooley, T., ‘The cubic case of the main conjecture in Vinogradov’s mean value theorem’, Adv. Math. 294 (2016), 532–561.Google Scholar | DOI

[70] Wooley, T. D., ‘Vinogradov’s mean value theorem via efficient congruencing’, Ann. Math. 175(3) (2012), 1575–1627.Google Scholar | DOI

[71] Ziegler, T., ‘Universal characteristic factors and Furstenberg averages’, J. Amer. Math. Soc. 20 (2007), 53–97.Google Scholar | DOI

[72] Zygmund, A., ‘An individual ergodic theorem for non-commutative transformations’, Acta Sci. Math. (Szeged) 14 (1951), 103–110.Google Scholar

[73] Zygmund, A., Trigonometric Series third edn. (Cambridge University Press, 2003).Google Scholar | DOI

Cité par Sources :