Uniformity in the Wiener-Wintner theorem for nilsequences

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August 2013, 33(8): 3497-3516. doi: 10.3934/dcds.2013.33.3497

Uniformity in the Wiener-Wintner theorem for nilsequences

Tanja Eisner ^1, and Pavel Zorin-Kranich ^2,

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

Received May 2012 Revised October 2012 Published January 2013

Abstract
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We prove a uniform extension of the Wiener-Wintner theorem for nilsequences due to Host and Kra and a nilsequence extension of the topological Wiener-Wintner theorem due to Assani. Our argument is based on (vertical) Fourier analysis and a Sobolev embedding theorem.

Keywords: Wiener-Wintner theorem, nilsequence, uniform convergence..

Mathematics Subject Classification: 37A45, 37A30, 28D0.

Citation: Tanja Eisner, Pavel Zorin-Kranich. Uniformity in the Wiener-Wintner theorem for nilsequences. Discrete and Continuous Dynamical Systems, 2013, 33 (8) : 3497-3516. doi: 10.3934/dcds.2013.33.3497

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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