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Uniformity in the Wiener-Wintner theorem for nilsequences
1. | KdV Institute for Mathematics, University of Amsterdam, P.O. Box 94248, 1090 GE Amsterdam |
2. | Korteweg-de Vries Institute for Mathematics, University of Amsterdam, P.O. Box 94248, 1090 GE Amsterdam, Netherlands |
References:
[1] |
Robert A. Adams and John J. F. Fournier, "Sobolev Spaces," Second edition, Pure and Applied Mathematics (Amsterdam), 140, Elsevier/Academic Press, Amsterdam, 2003. |
[2] |
Idris Assani and Kimberly Presser, Pointwise characteristic factors for the multiterm return times theorem, Ergodic Theory Dynam. Systems, 32 (2012), 341-360. |
[3] |
Idris Assani, "Wiener Wintner Ergodic Theorems," World Scientific Publishing Co., Inc., River Edge, NJ, 2003. |
[4] |
Idris Assani, Pointwise convergence of ergodic averages along cubes, J. Anal. Math., 110 (2010), 241-269.
doi: 10.1007/s11854-010-0006-3. |
[5] |
Vitaly Bergelson and Alexander Leibman, Distribution of values of bounded generalized polynomials, Acta Math., 198 (2007), 155-230.
doi: 10.1007/s11511-007-0015-y. |
[6] |
J. Bourgain, Double recurrence and almost sure convergence, J. Reine Angew. Math., 404 (1990), 140-161.
doi: 10.1515/crll.1990.404.140. |
[7] |
S. Butkevich, "Convergence of Averages in Ergodic Theory," Ph.D. thesis, Ohio State University, 2000. |
[8] |
Qing Chu, Nikos Frantzikinakis and Bernard Host, Ergodic averages of commuting transformations with distinct degree polynomial iterates, Proc. Lond. Math. Soc., 102 (2011), 801-842.
doi: 10.1112/plms/pdq037. |
[9] |
Qing Chu, Convergence of weighted polynomial multiple ergodic averages, Proc. Amer. Math. Soc., 137 (2009), 1363-1369.
doi: 10.1090/S0002-9939-08-09614-7. |
[10] |
Andrés del Junco and Joseph Rosenblatt, Counterexamples in ergodic theory and number theory, Math. Ann., 245 (1979), 185-197.
doi: 10.1007/BF01673506. |
[11] |
Tanja Eisner and Terence Tao, Large values of the Gowers-Host-Kra seminorms, J. Anal. Math., 117 (2012), 133-186.
doi: 10.1007/s11854-012-0018-2. |
[12] |
Nikos Frantzikinakis, Uniformity in the polynomial Wiener-Wintner theorem, Ergodic Theory Dynam. Systems, 26 (2006), 1061-1071.
doi: 10.1017/S0143385706000204. |
[13] |
H. Furstenberg, "Recurrence in Ergodic Theory and Combinatorial Number Theory," M. B. Porter Lectures, Princeton University Press, Princeton, N.J., 1981. |
[14] |
Benjamin Green and Terence Tao, Linear equations in primes, Ann. of Math., 171 (2010), 1753-1850.
doi: 10.4007/annals.2010.171.1753. |
[15] |
Ben Green and Terence Tao, The quantitative behaviour of polynomial orbits on nilmanifolds, Ann. of Math., 175 (2012), 465-540.
doi: 10.4007/annals.2012.175.2.2. |
[16] |
Ben Green, Terence Tao and Tamar Ziegler, An inverse theorem for the Gowers $U^{s+1}[N]$-norm, Ann. of Math., 176 (2012), 1231-1372.
doi: 10.4007/annals.2012.176.2.11. |
[17] |
Bernard Host and Bryna Kra, Nonconventional ergodic averages and nilmanifolds, Ann. of Math., 161 (2005), 397-488.
doi: 10.4007/annals.2005.161.397. |
[18] |
Bernard Host and Bryna Kra, Analysis of two step nilsequences, Ann. Inst. Fourier (Grenoble), 58 (2008), 1407-1453. |
[19] |
Bernard Host and Bryna Kra, Uniformity seminorms on $l^\infty$ and applications, J. Anal. Math., 108 (2009), 219-276.
doi: 10.1007/s11854-009-0024-1. |
[20] |
Bernard Host, Bryna Kra and Alejandro Maass, Nilsequences and a structure theorem for topological dynamical systems, Adv. Math., 224 (2010), 103-129.
doi: 10.1016/j.aim.2009.11.009. |
[21] |
Bernard Host, Bryna Kra and Alejandro Maass, Complexity of nilsystems and systems lacking nilfactors, preprint, (2012), arXiv:1203.3778. |
[22] |
Jean-Pierre Kahane, "Some Random Series of Functions," Second edition, Cambridge Studies in Advanced Mathematics, 5, Cambridge University Press, Cambridge, 1985. |
[23] |
A. Leibman, Polynomial mappings of groups, Israel J. Math., 129 (2002), 29-60. See http://www.math.osu.edu/ leibman.1/preprints/PolMapG-err.pdf for erratum.
doi: 10.1007/BF02773152. |
[24] |
A. Leibman, Convergence of multiple ergodic averages along polynomials of several variables, Israel J. Math., 146 (2005), 303-315.
doi: 10.1007/BF02773538. |
[25] |
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. |
[26] |
Daniel Lenz, Continuity of eigenfunctions of uniquely ergodic dynamical systems and intensity of Bragg peaks, Comm. Math. Phys., 287 (2009), 225-258.
doi: 10.1007/s00220-008-0594-2. |
[27] |
E. Lesigne, Un théorème de disjonction de systèmes dynamiques et une généralisation du théorème ergodique de Wiener-Wintner, Ergodic Theory Dynam. Systems, 10 (1990), 513-521.
doi: 10.1017/S014338570000571X. |
[28] |
E. Lesigne, Spectre quasi-discret et théorème ergodique de Wiener-Wintner pour les polynômes, Ergodic Theory Dynam. Systems, 13 (1993), 767-784. |
[29] |
Elon Lindenstrauss, Pointwise theorems for amenable groups, Invent. Math., 146 (2001), 259-295.
doi: 10.1007/s002220100162. |
[30] |
A. I. Mal'cev, On a class of homogeneous spaces, Izvestiya Akad. Nauk. SSSR. Ser. Mat., 13 (1949), 9-32. |
[31] |
E. Arthur Robinson, Jr., On uniform convergence in the Wiener-Wintner theorem, J. London Math. Soc., 49 (1994), 493-501.
doi: 10.1112/jlms/49.3.493. |
[32] |
Joseph M. Rosenblatt and Máté Wierdl, A new maximal inequality and its applications, Ergodic Theory Dynam. Systems, 12 (1992), 509-558.
doi: 10.1017/S0143385700006921. |
[33] |
Terence Tao, "Higher Order Fourier Analysis," Graduate Studies in Mathematics, 142, American Mathematical Society, Providence, RI, 2012. |
[34] |
Peter Walters, "An Introduction to Ergodic Theory," Graduate Texts in Mathematics, 79, Springer-Verlag, New York-Berlin, 1982. |
[35] |
Norbert Wiener and Aurel Wintner, Harmonic analysis and ergodic theory, Amer. J. Math., 63 (1941), 415-426. |
[36] |
Pavel Zorin-Kranich, A nilpotent IP polynomial multiple recurrence theorem, preprint, 2012, arXiv:1206.0287. |
show all references
References:
[1] |
Robert A. Adams and John J. F. Fournier, "Sobolev Spaces," Second edition, Pure and Applied Mathematics (Amsterdam), 140, Elsevier/Academic Press, Amsterdam, 2003. |
[2] |
Idris Assani and Kimberly Presser, Pointwise characteristic factors for the multiterm return times theorem, Ergodic Theory Dynam. Systems, 32 (2012), 341-360. |
[3] |
Idris Assani, "Wiener Wintner Ergodic Theorems," World Scientific Publishing Co., Inc., River Edge, NJ, 2003. |
[4] |
Idris Assani, Pointwise convergence of ergodic averages along cubes, J. Anal. Math., 110 (2010), 241-269.
doi: 10.1007/s11854-010-0006-3. |
[5] |
Vitaly Bergelson and Alexander Leibman, Distribution of values of bounded generalized polynomials, Acta Math., 198 (2007), 155-230.
doi: 10.1007/s11511-007-0015-y. |
[6] |
J. Bourgain, Double recurrence and almost sure convergence, J. Reine Angew. Math., 404 (1990), 140-161.
doi: 10.1515/crll.1990.404.140. |
[7] |
S. Butkevich, "Convergence of Averages in Ergodic Theory," Ph.D. thesis, Ohio State University, 2000. |
[8] |
Qing Chu, Nikos Frantzikinakis and Bernard Host, Ergodic averages of commuting transformations with distinct degree polynomial iterates, Proc. Lond. Math. Soc., 102 (2011), 801-842.
doi: 10.1112/plms/pdq037. |
[9] |
Qing Chu, Convergence of weighted polynomial multiple ergodic averages, Proc. Amer. Math. Soc., 137 (2009), 1363-1369.
doi: 10.1090/S0002-9939-08-09614-7. |
[10] |
Andrés del Junco and Joseph Rosenblatt, Counterexamples in ergodic theory and number theory, Math. Ann., 245 (1979), 185-197.
doi: 10.1007/BF01673506. |
[11] |
Tanja Eisner and Terence Tao, Large values of the Gowers-Host-Kra seminorms, J. Anal. Math., 117 (2012), 133-186.
doi: 10.1007/s11854-012-0018-2. |
[12] |
Nikos Frantzikinakis, Uniformity in the polynomial Wiener-Wintner theorem, Ergodic Theory Dynam. Systems, 26 (2006), 1061-1071.
doi: 10.1017/S0143385706000204. |
[13] |
H. Furstenberg, "Recurrence in Ergodic Theory and Combinatorial Number Theory," M. B. Porter Lectures, Princeton University Press, Princeton, N.J., 1981. |
[14] |
Benjamin Green and Terence Tao, Linear equations in primes, Ann. of Math., 171 (2010), 1753-1850.
doi: 10.4007/annals.2010.171.1753. |
[15] |
Ben Green and Terence Tao, The quantitative behaviour of polynomial orbits on nilmanifolds, Ann. of Math., 175 (2012), 465-540.
doi: 10.4007/annals.2012.175.2.2. |
[16] |
Ben Green, Terence Tao and Tamar Ziegler, An inverse theorem for the Gowers $U^{s+1}[N]$-norm, Ann. of Math., 176 (2012), 1231-1372.
doi: 10.4007/annals.2012.176.2.11. |
[17] |
Bernard Host and Bryna Kra, Nonconventional ergodic averages and nilmanifolds, Ann. of Math., 161 (2005), 397-488.
doi: 10.4007/annals.2005.161.397. |
[18] |
Bernard Host and Bryna Kra, Analysis of two step nilsequences, Ann. Inst. Fourier (Grenoble), 58 (2008), 1407-1453. |
[19] |
Bernard Host and Bryna Kra, Uniformity seminorms on $l^\infty$ and applications, J. Anal. Math., 108 (2009), 219-276.
doi: 10.1007/s11854-009-0024-1. |
[20] |
Bernard Host, Bryna Kra and Alejandro Maass, Nilsequences and a structure theorem for topological dynamical systems, Adv. Math., 224 (2010), 103-129.
doi: 10.1016/j.aim.2009.11.009. |
[21] |
Bernard Host, Bryna Kra and Alejandro Maass, Complexity of nilsystems and systems lacking nilfactors, preprint, (2012), arXiv:1203.3778. |
[22] |
Jean-Pierre Kahane, "Some Random Series of Functions," Second edition, Cambridge Studies in Advanced Mathematics, 5, Cambridge University Press, Cambridge, 1985. |
[23] |
A. Leibman, Polynomial mappings of groups, Israel J. Math., 129 (2002), 29-60. See http://www.math.osu.edu/ leibman.1/preprints/PolMapG-err.pdf for erratum.
doi: 10.1007/BF02773152. |
[24] |
A. Leibman, Convergence of multiple ergodic averages along polynomials of several variables, Israel J. Math., 146 (2005), 303-315.
doi: 10.1007/BF02773538. |
[25] |
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. |
[26] |
Daniel Lenz, Continuity of eigenfunctions of uniquely ergodic dynamical systems and intensity of Bragg peaks, Comm. Math. Phys., 287 (2009), 225-258.
doi: 10.1007/s00220-008-0594-2. |
[27] |
E. Lesigne, Un théorème de disjonction de systèmes dynamiques et une généralisation du théorème ergodique de Wiener-Wintner, Ergodic Theory Dynam. Systems, 10 (1990), 513-521.
doi: 10.1017/S014338570000571X. |
[28] |
E. Lesigne, Spectre quasi-discret et théorème ergodique de Wiener-Wintner pour les polynômes, Ergodic Theory Dynam. Systems, 13 (1993), 767-784. |
[29] |
Elon Lindenstrauss, Pointwise theorems for amenable groups, Invent. Math., 146 (2001), 259-295.
doi: 10.1007/s002220100162. |
[30] |
A. I. Mal'cev, On a class of homogeneous spaces, Izvestiya Akad. Nauk. SSSR. Ser. Mat., 13 (1949), 9-32. |
[31] |
E. Arthur Robinson, Jr., On uniform convergence in the Wiener-Wintner theorem, J. London Math. Soc., 49 (1994), 493-501.
doi: 10.1112/jlms/49.3.493. |
[32] |
Joseph M. Rosenblatt and Máté Wierdl, A new maximal inequality and its applications, Ergodic Theory Dynam. Systems, 12 (1992), 509-558.
doi: 10.1017/S0143385700006921. |
[33] |
Terence Tao, "Higher Order Fourier Analysis," Graduate Studies in Mathematics, 142, American Mathematical Society, Providence, RI, 2012. |
[34] |
Peter Walters, "An Introduction to Ergodic Theory," Graduate Texts in Mathematics, 79, Springer-Verlag, New York-Berlin, 1982. |
[35] |
Norbert Wiener and Aurel Wintner, Harmonic analysis and ergodic theory, Amer. J. Math., 63 (1941), 415-426. |
[36] |
Pavel Zorin-Kranich, A nilpotent IP polynomial multiple recurrence theorem, preprint, 2012, arXiv:1206.0287. |
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