×

Pointwise ergodic theorems for non-conventional bilinear polynomial averages. (English) Zbl 1505.37010

The authors establish convergence in norm and pointwise almost everywhere for the non-conventional bilinear polynomial ergodic averages \[\frac1n \sum_{k=1}^n f(T^k x) g(T^{p(k)}x)\] for a \(\sigma\)-finite measure-preserving map \(T\) and a polynomial \(p(k)\) of degree \(d\ge2\). The methods combine techniques from harmonic analysis with the recent inverse theorems of S. Peluse and S. Prendiville [“Quantitative bounds in the nonlinear Roth theorem”, Preprint, arXiv:1903.02592] in additive combinatorics. At large scales, the harmonic analysis of the adelic integers plays a role too.

MSC:

37A30 Ergodic theorems, spectral theory, Markov operators
37A44 Relations between ergodic theory and number theory
37A46 Relations between ergodic theory and harmonic analysis
42A45 Multipliers in one variable harmonic analysis
42A50 Conjugate functions, conjugate series, singular integrals
42A85 Convolution, factorization for one variable harmonic analysis
43A25 Fourier and Fourier-Stieltjes transforms on locally compact and other abelian groups
11L03 Trigonometric and exponential sums (general theory)
11L07 Estimates on exponential sums
11L15 Weyl sums
11P55 Applications of the Hardy-Littlewood method
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] {\relax el Abdalaoui}, {\relax el H}, Simple proof of {B}ourgain bilinear ergodic theorem and its extension to polynomials and polynomials in primes (2019)
[2] Austin, Tim, A proof of {W}alsh’s convergence theorem using couplings, Int. Math. Res. Not. IMRN. International Mathematics Research Notices. IMRN, 6661-6674 (2015) · Zbl 1372.37012 · doi:10.1093/imrn/rnu145
[3] Bergelson, Vitaly, Weakly mixing {PET}, Ergodic Theory Dynam. Systems. Ergodic Theory and Dynamical Systems, 7, 337-349 (1987) · Zbl 0645.28012 · doi:10.1017/S0143385700004090
[4] Bergelson, Vitaly, Ergodic {R}amsey theory-an update. Ergodic Theory of {\({\bf Z}^d\)} Actions, London Math. Soc. Lecture Note Ser., 228, 1-61 (1996) · Zbl 0846.05095 · doi:10.1017/CBO9780511662812.002
[5] Bergelson, Vitaly, Combinatorial and {D}iophantine applications of ergodic theory. Handbook of Dynamical Systems. {V}ol. 1{B}, 745-869 (2006) · Zbl 1130.37317 · doi:10.1016/S1874-575X(06)80037-8
[6] Bergelson, Vitaly; Leibman, A., A nilpotent {R}oth theorem, Invent. Math.. Inventiones Mathematicae, 147, 429-470 (2002) · Zbl 1042.37001 · doi:10.1007/s002220100179
[7] Bergelson, Vitaly; Leibman, A., Polynomial extensions of van der {W}aerden’s and {S}zemer\'{e}di’s theorems, J. Amer. Math. Soc.. Journal of the Amer. Math. Soc., 9, 725-753 (1996) · Zbl 0870.11015 · doi:10.1090/S0894-0347-96-00194-4
[8] Birkhoff, G. D., Proof of the ergodic theorem, Proc. Natl. Acad. Sci. USA, 17, 656-660 (1931) · JFM 57.1011.02 · doi:10.1073/pnas.17.2.656
[9] Bourgain, Jean, A nonlinear version of {R}oth’s theorem for sets of positive density in the real line, J. Analyse Math.. Journal d’Analyse Math\'{e}matique, 50, 169-181 (1988) · Zbl 0675.42010 · doi:10.1007/BF02796120
[10] Bourgain, Jean, On the maximal ergodic theorem for certain subsets of the integers, Israel J. Math.. Israel Journal of Mathematics, 61, 39-72 (1988) · Zbl 0642.28010 · doi:10.1007/BF02776301
[11] Bourgain, Jean, On the pointwise ergodic theorem on {\(L^p\)} for arithmetic sets, Israel J. Math.. Israel Journal of Mathematics, 61, 73-84 (1988) · Zbl 0642.28011 · doi:10.1007/BF02776302
[12] with an appendix by the author, Harry Furstenberg, Yitzhak Katznelson; Donald S. Ornstein, Pointwise ergodic theorems for arithmetic sets, Inst. Hautes \'{E}tudes Sci. Publ. Math.. Institut des Hautes \'{E}tudes Scientifiques. Publications Math\'{e}matiques, 5-45 (1989) · Zbl 0705.28008
[13] Bourgain, Jean, Double recurrence and almost sure convergence, J. Reine Angew. Math.. Journal f\"{u}r die Reine und Angewandte Mathematik. [Crelle’s Journal], 404, 140-161 (1990) · Zbl 0685.28008 · doi:10.1515/crll.1990.404.140
[14] Bourgain, Jean; Chang, M.-C., Nonlinear {R}oth type theorems in finite fields, Israel J. Math.. Israel Journal of Mathematics, 221, 853-867 (2017) · Zbl 1420.11024 · doi:10.1007/s11856-017-1577-9
[15] Boshernitzan, Michael; Kolesnik, Grigori; Quas, Anthony; Wierdl, M\'{a}t\'{e}, Ergodic averaging sequences, J. Anal. Math.. Journal d’Analyse Math\'{e}matique, 95, 63-103 (2005) · Zbl 1100.28010 · doi:10.1007/BF02791497
[16] Boshernitzan, Michael; Wierdl, M\'{a}t\'{e}, Ergodic theorems along sequences and {H}ardy fields, Proc. Nat. Acad. Sci. U.S.A.. Proceedings of the National Academy of Sciences of the United States of America, 93, 8205-8207 (1996) · Zbl 0863.28011 · doi:10.1073/pnas.93.16.8205
[17] Bruhat, Fran\c{c}ois, Distributions sur un groupe localement compact et applications \`a l’\'{e}tude des repr\'{e}sentations des groupes {\( \wp \)}-adiques, Bull. Soc. Math. France. Bulletin de la Soci\'{e}t\'{e} Math\'{e}matique de France, 89, 43-75 (1961) · Zbl 0128.35701 · doi:10.24033/bsmf.1559
[18] Buczolich, Zolt\'{a}n; Mauldin, R. Daniel, Divergent square averages, Ann. of Math. (2). Annals of Mathematics. Second Series, 171, 1479-1530 (2010) · Zbl 1200.37003 · doi:10.4007/annals.2010.171.1479
[19] Calder\'{o}n, A.-P., Ergodic theory and translation-invariant operators, Proc. Nat. Acad. Sci. U.S.A.. Proceedings of the National Academy of Sciences of the United States of America, 59, 349-353 (1968) · Zbl 0185.21806 · doi:10.1073/pnas.59.2.349
[20] Chu, Qing; Frantzikinakis, Nikos; Host, Bernard, Ergodic averages of commuting transformations with distinct degree polynomial iterates, Proc. Lond. Math. Soc. (3). Proceedings of the London Mathematical Society. Third Series, 102, 801-842 (2011) · Zbl 1218.37009 · doi:10.1112/plms/pdq037
[21] Dasu, Shival; Demeter, Ciprian; Langowski, Bartosz, Sharp {\( \ell^p\)}-improving estimates for the discrete paraboloid, J. Fourier Anal. Appl.. The Journal of Fourier Analysis and Applications, 27, Paper No. 3 pp. (2021) · Zbl 1457.42013 · doi:10.1007/s00041-020-09801-2
[22] Demeter, Ciprian, Pointwise convergence of the ergodic bilinear {H}ilbert transform, Illinois J. Math.. Illinois Journal of Mathematics, 51, 1123-1158 (2007) · Zbl 1175.37013 · doi:10.1215/ijm/1258138536
[23] Derrien, Jean-Marc; Lesigne, Emmanuel, Un th\'{e}or\`eme ergodique polynomial ponctuel pour les endomorphismes exacts et les {\(K\)}-syst\`emes, Ann. Inst. H. Poincar\'{e} Probab. Statist.. Annales de l’Institut Henri Poincar\'{e}. Probabilit\'{e}s et Statistiques, 32, 765-778 (1996) · Zbl 0868.60028
[24] Demeter, Ciprian; Lacey, Michael T.; Tao, Terence; Thiele, Christoph, Breaking the duality in the return times theorem, Duke Math. J.. Duke Mathematical Journal, 143, 281-355 (2008) · Zbl 1213.42064 · doi:10.1215/00127094-2008-020
[25] Demeter, Ciprian; Tao, Terence; Thiele, Christoph, Maximal multilinear operators, Trans. Amer. Math. Soc.. Transactions of the Amer. Math. Soc., 360, 4989-5042 (2008) · Zbl 1268.42034 · doi:10.1090/S0002-9947-08-04474-7
[26] Do, Yen; Muscalu, Camil; Thiele, Christoph, Variational estimates for paraproducts, Rev. Mat. Iberoam.. Revista Matem\'{a}tica Iberoamericana, 28, 857-878 (2012) · Zbl 1256.42018 · doi:10.4171/RMI/694
[27] Dong, Dong, On a discrete bilinear singular operator, C. R. Math. Acad. Sci. Paris. Comptes Rendus Math\'{e}matique. Acad\'{e}mie des Sciences. Paris, 355, 538-542 (2017) · Zbl 1365.42009 · doi:10.1016/j.crma.2017.03.010
[28] Dong, Dong; Li, Xiaochun; Sawin, Will, Improved estimates for polynomial {R}oth type theorems in finite fields, J. Anal. Math.. Journal d’Analyse Math\'{e}matique, 141, 689-705 (2020) · Zbl 1475.11022 · doi:10.1007/s11854-020-0113-8
[29] Durcik, Polona; Guo, Shaoming; Roos, Joris, A polynomial {R}oth theorem on the real line, Trans. Amer. Math. Soc.. Transactions of the Amer. Math. Soc., 371, 6973-6993 (2019) · Zbl 1432.42005 · doi:10.1090/tran/7574
[30] Frantzikinakis, Nikos, Some open problems on multiple ergodic averages, Bull. Hellenic Math. Soc.. Bulletin of the Hellenic Mathematical Society, 60, 41-90 (2016) · Zbl 1425.37004
[31] Frantzikinakis, Nikos; Kra, Bryna, Polynomial averages converge to the product of integrals. {P}robability in mathematics, Israel J. Math.. Israel Journal of Mathematics, 148, 267-276 (2005) · Zbl 1155.37303 · doi:10.1007/BF02775439
[32] Frantzikinakis, Nikos; Lesigne, Emmanuel; Wierdl, M\'{a}t\'{e}, Random differences in {S}zemer\'{e}di’s theorem and related results, J. Anal. Math.. Journal d’Analyse Math\'{e}matique, 130, 91-133 (2016) · Zbl 1414.11018 · doi:10.1007/s11854-016-0030-z
[33] Furstenberg, Harry, Ergodic behavior of diagonal measures and a theorem of {S}zemer\'{e}di on arithmetic progressions, J. Analyse Math.. Journal d’Analyse Math\'{e}matique, 31, 204-256 (1977) · Zbl 0347.28016 · doi:10.1007/BF02813304
[34] Furstenberg, Hillel, Nonconventional ergodic averages. The Legacy of {J}ohn von {N}eumann, Proc. Sympos. Pure Math., 50, 43-56 (1990) · Zbl 0711.28006 · doi:10.1090/pspum/050/1067751
[35] Furstenberg, Hillel, Recurrence in Ergodic Theory and Combinatorial Number Theory, xi+203 pp. (1981) · Zbl 0459.28023 · doi:10.1515/9781400855162
[36] Furstenberg, Hillel; Weiss, Benjamin, A mean ergodic theorem for {\((1/N)\sum^N_{n=1}f(T^nx)g(T^{n^2}x)\)}. Convergence in Ergodic Theory and Probability, Ohio State Univ. Math. Res. Inst. Publ., 5, 193-227 (1996) · Zbl 0869.28010 · doi:10.1515/9783110889383.193
[37] Gaitan, Alejandra; Lie, Victor, The boundedness of the (sub)bilinear maximal function along “non-flat” smooth curves, J. Fourier Anal. Appl.. The Journal of Fourier Analysis and Applications, 26, 69-33 (2020) · Zbl 1447.42018 · doi:10.1007/s00041-020-09770-6
[38] Gowers, W. T., Decompositions, approximate structure, transference, and the {H}ahn-{B}anach theorem, Bull. Lond. Math. Soc.. Bulletin of the London Mathematical Society, 42, 573-606 (2010) · Zbl 1233.05198 · doi:10.1112/blms/bdq018
[39] Gowers, W. T.; Wolf, J., The true complexity of a system of linear equations, Proc. Lond. Math. Soc. (3). Proceedings of the London Mathematical Society. Third Series, 100, 155-176 (2010) · Zbl 1243.11010 · doi:10.1112/plms/pdp019
[40] Green, Ben; Tao, Terence, Quadratic uniformity of the {M}\"{o}bius function, Ann. Inst. Fourier (Grenoble). Universit\'{e} de Grenoble. Annales de l’Institut Fourier, 58, 1863-1935 (2008) · Zbl 1160.11017 · doi:10.5802/aif.2401
[41] Han, Rui; Kova\v{c}, Vjekoslav; Lacey, Michael T.; Madrid, Jos\'{e}; Yang, Fan, Improving estimates for discrete polynomial averages, J. Fourier Anal. Appl.. The Journal of Fourier Analysis and Applications, 26, 42-11 (2020) · Zbl 1442.42024 · doi:10.1007/s00041-020-09748-4
[42] Hopf, Eberhard, The general temporally discrete {M}arkoff process, J. Rational Mech. Anal.. Journal of Rational Mechanics and Analysis, 3, 13-45 (1954) · Zbl 0055.36705 · doi:10.1512/iumj.1954.3.53002
[43] Host, Bernard; Kra, Bryna, Nonconventional ergodic averages and nilmanifolds, Ann. of Math. (2). Annals of Mathematics. Second Series, 161, 397-488 (2005) · Zbl 1077.37002 · doi:10.4007/annals.2005.161.397
[44] Host, Bernard; Kra, Bryna, Convergence of polynomial ergodic averages. {P}robability in mathematics, Israel J. Math.. Israel Journal of Mathematics, 149, 1-19 (2005) · Zbl 1085.28009 · doi:10.1007/BF02772534
[45] Host, Bernard; Kra, Bryna, A point of view on {G}owers uniformity norms, New York J. Math.. New York Journal of Mathematics, 18, 213-248 (2012) · Zbl 1268.11018
[46] Ionescu, Alexandru D.; Wainger, Stephen, {\(L^p\)} boundedness of discrete singular {R}adon transforms, J. Amer. Math. Soc.. Journal of the Amer. Math. Soc., 19, 357-383 (2006) · Zbl 1158.42007 · doi:10.1090/S0894-0347-05-00508-4
[47] Iwaniec, Henryk; Kowalski, Emmanuel, Analytic Number Theory, Amer. Math. Soc. Colloq. Publ., 53, xii+615 pp. (2004) · Zbl 1059.11001 · doi:10.1090/coll/053
[48] Jones, Roger L.; Kaufman, Robert; Rosenblatt, Joseph M.; Wierdl, M\'{a}t\'{e}, Oscillation in ergodic theory, Ergodic Theory Dynam. Systems. Ergodic Theory and Dynamical Systems, 18, 889-935 (1998) · Zbl 0924.28009 · doi:10.1017/S0143385798108349
[49] Jones, Roger L.; Seeger, Andreas; Wright, James, Strong variational and jump inequalities in harmonic analysis, Trans. Amer. Math. Soc.. Transactions of the Amer. Math. Soc., 360, 6711-6742 (2008) · Zbl 1159.42013 · doi:10.1090/S0002-9947-08-04538-8
[50] Jones, Roger L.; Wang, Gang, Variation inequalities for the {F}ej\'{e}r and {P}oisson kernels, Trans. Amer. Math. Soc.. Transactions of the Amer. Math. Soc., 356, 4493-4518 (2004) · Zbl 1065.42006 · doi:10.1090/S0002-9947-04-03397-5
[51] Kowalski, Michael W.; Wright, James, Elementary inequalities involving the roots of a polynomial with applications in harmonic analysis and number theory, J. Lond. Math. Soc. (2). Journal of the London Mathematical Society. Second Series, 86, 835-851 (2012) · Zbl 1291.11109 · doi:10.1112/jlms/jds029
[52] Kra, B., Private communication, {O}ctober (2021) (2021)
[53] Krause, B., Polynomial Ergodic Averages Converge Rapidly: {V}ariations on a Theorem of {B}ourgain (2014)
[54] Lacey, Michael T., The bilinear maximal functions map into {\(L^p\)} for {\(2/3<p\leq1\)}, Ann. of Math. (2). Annals of Mathematics. Second Series, 151, 35-57 (2000) · Zbl 0967.47031 · doi:10.2307/121111
[55] LaVictoire, Patrick, Universally {\(L^1\)}-bad arithmetic sequences, J. Anal. Math.. Journal d’Analyse Math\'{e}matique, 113, 241-263 (2011) · Zbl 1220.37005 · doi:10.1007/s11854-011-0006-y
[56] Leibman, A., Convergence of multiple ergodic averages along polynomials of several variables, Israel J. Math.. Israel Journal of Mathematics, 146, 303-315 (2005) · Zbl 1080.37002 · doi:10.1007/BF02773538
[57] Leibman, A., Pointwise convergence of ergodic averages for polynomial sequences of translations on a nilmanifold, Ergodic Theory Dynam. Systems. Ergodic Theory and Dynamical Systems, 25, 201-213 (2005) · Zbl 1080.37003 · doi:10.1017/S0143385704000215
[58] Lewko, Allison; Lewko, Mark, Estimates for the square variation of partial sums of {F}ourier series and their rearrangements, J. Funct. Anal.. Journal of Functional Analysis, 262, 2561-2607 (2012) · Zbl 1239.42027 · doi:10.1016/j.jfa.2011.12.007
[59] Li, Xiaochun, Bilinear {H}ilbert transforms along curves {I}: {T}he monomial case, Anal. PDE. Analysis & PDE, 6, 197-220 (2013) · Zbl 1275.42022 · doi:10.2140/apde.2013.6.197
[60] Li, Xiaochun; Xiao, Lechao, Uniform estimates for bilinear {H}ilbert transforms and bilinear maximal functions associated to polynomials, Amer. J. Math.. American Journal of Mathematics, 138, 907-962 (2016) · Zbl 1355.42011 · doi:10.1353/ajm.2016.0030
[61] Lie, Victor, On the boundedness of the bilinear {H}ilbert transform along “non-flat” smooth curves, Amer. J. Math.. American Journal of Mathematics, 137, 313-363 (2015) · Zbl 1318.42008 · doi:10.1353/ajm.2015.0013
[62] Lie, Victor, On the boundedness of the bilinear {H}ilbert transform along “non-flat” smooth curves. {T}he {B}anach triangle case {\((L^r, 1\leq r<\infty)\)}, Rev. Mat. Iberoam.. Revista Matem\'{a}tica Iberoamericana, 34, 331-353 (2018) · Zbl 1395.42009 · doi:10.4171/RMI/987
[63] Magyar, {\'A}.; Stein, E. M.; Wainger, S., Discrete analogues in harmonic analysis: spherical averages, Ann. of Math. (2). Annals of Mathematics. Second Series, 155, 189-208 (2002) · Zbl 1036.42018 · doi:10.2307/3062154
[64] Mirek, Mariusz, Square function estimates for discrete {R}adon transforms, Anal. PDE. Analysis & PDE, 11, 583-608 (2018) · Zbl 1383.42014 · doi:10.2140/apde.2018.11.583
[65] Ionescu, A. D.; Magyar, {\'{A}}.; Mirek, M.; Szarek, T. Z., Polynomial averages and pointwise ergodic theorems on nilpotent groups (2021)
[66] Mirek, Mariusz; Stein, Elias M.; Trojan, Bartosz, {\( \ell^p(\Bbb Z^d) \)}-estimates for discrete operators of {R}adon type: variational estimates, Invent. Math.. Inventiones Mathematicae, 209, 665-748 (2017) · Zbl 1397.42006 · doi:10.1007/s00222-017-0718-4
[67] Mirek, Mariusz; Stein, Elias M.; Zorin-Kranich, Pavel, Jump inequalities via real interpolation, Math. Ann.. Mathematische Annalen, 376, 797-819 (2020) · Zbl 1448.60052 · doi:10.1007/s00208-019-01889-2
[68] Mirek, Mariusz; Stein, Elias M.; Zorin-Kranich, Pavel, A bootstrapping approach to jump inequalities and their applications, Anal. PDE. Analysis & PDE, 13, 527-558 (2020) · Zbl 1437.42023 · doi:10.2140/apde.2020.13.527
[69] Mirek, Mariusz; Stein, Elias M.; Zorin-Kranich, Pavel, Jump inequalities for translation-invariant operators of {R}adon type on {\( \Bbb Z^d\)}, Adv. Math.. Advances in Mathematics, 365, 107065-57 (2020) · Zbl 1436.42022 · doi:10.1016/j.aim.2020.107065
[70] von Neumann, J., Proof of the quasi-ergodic hypothesis, Proc. Natl. Acad. Sci. USA, 18, 70-82 (1932) · JFM 58.1271.03 · doi:10.1073/pnas.18.1.70
[71] Osborne, M. Scott, On the {S}chwartz-{B}ruhat space and the {P}aley-{W}iener theorem for locally compact abelian groups, J. Functional Analysis, 19, 40-49 (1975) · Zbl 0295.43008 · doi:10.1016/0022-1236(75)90005-1
[72] Peluse, Sarah, Three-term polynomial progressions in subsets of finite fields, Israel J. Math.. Israel Journal of Mathematics, 228, 379-405 (2018) · Zbl 1456.11015 · doi:10.1007/s11856-018-1768-z
[73] Peluse, Sarah, Bounds for {S}ets with {N}o {N}ontrivial {P}olynomial {P}rogressions, 209 pp. (2019)
[74] Peluse, Sarah; Prendiville, S., Quantitative bounds in the non-linear {R}oth theorem (2019)
[75] Peluse, Sarah; Prendiville, S., A polylogarithmic bound in the nonlinear {R}oth theorem, Int. Math. Res. Not. IMRN. International Mathematics Research Notices. IMRN (2020) · Zbl 1508.11042 · doi:10.1093/imrn/rnaa261
[76] Pierce, L. B., On superorthogonality, J. Geom. Anal.. Journal of Geometric Analysis, 31, 7096-7183 (2021) · Zbl 1471.42037 · doi:10.1007/s12220-021-00606-3
[77] Prendiville, S., The inverse theorem for the nonlinear {R}oth configuration: an exposition (2020)
[78] Rudin, Walter, Fourier Analysis on Groups, Interscience Tracts Pure Appl. Math., 12, ix+285 pp. (1962) · Zbl 0698.43001 · doi:10.1002/9781118165621
[79] Szemer\'{e}di, E., On sets of integers containing no {\(k\)} elements in arithmetic progression, Acta Arith.. Polska Akademia Nauk. Instytut Matematyczny. Acta Arithmetica, 27, 199-245 (1975) · Zbl 0303.10056 · doi:10.4064/aa-27-1-199-245
[80] Tao, Terence, The {I}onescu-{W}ainger multiplier theorem and the adeles, Mathematika. Mathematika. A Journal of Pure and Applied Mathematics, 67, 647-677 (2021) · Zbl 1523.42017 · doi:10.1112/mtk.12094
[81] Tate, J. T., Fourier analysis in number fields, and {H}ecke’s zeta-functions. Algebraic {N}umber {T}heory, 305-347 (1967) · Zbl 1492.11154
[82] Walsh, Miguel N., Norm convergence of nilpotent ergodic averages, Ann. of Math. (2). Annals of Mathematics. Second Series, 175, 1667-1688 (2012) · Zbl 1248.37008 · doi:10.4007/annals.2012.175.3.15
[83] Ziegler, Tamar, Universal characteristic factors and {F}urstenberg averages, J. Amer. Math. Soc.. Journal of the Amer. Math. Soc., 20, 53-97 (2007) · Zbl 1198.37014 · doi:10.1090/S0894-0347-06-00532-7
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.