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 & Continuous Dynamical Systems - A, 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, 140 (2003).
[2]	Idris Assani and Kimberly Presser, Pointwise characteristic factors for the multiterm return times theorem,, Ergodic Theory Dynam. Systems, 32 (2012), 341.
[3]	Idris Assani, "Wiener Wintner Ergodic Theorems,", World Scientific Publishing Co., (2003).
[4]	Idris Assani, Pointwise convergence of ergodic averages along cubes,, J. Anal. Math., 110 (2010), 241. 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. doi: 10.1007/s11511-007-0015-y.
[6]	J. Bourgain, Double recurrence and almost sure convergence,, J. Reine Angew. Math., 404 (1990), 140. doi: 10.1515/crll.1990.404.140.
[7]	S. Butkevich, "Convergence of Averages in Ergodic Theory,", Ph.D. thesis, (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. doi: 10.1112/plms/pdq037.
[9]	Qing Chu, Convergence of weighted polynomial multiple ergodic averages,, Proc. Amer. Math. Soc., 137 (2009), 1363. 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. doi: 10.1007/BF01673506.
[11]	Tanja Eisner and Terence Tao, Large values of the Gowers-Host-Kra seminorms,, J. Anal. Math., 117 (2012), 133. doi: 10.1007/s11854-012-0018-2.
[12]	Nikos Frantzikinakis, Uniformity in the polynomial Wiener-Wintner theorem,, Ergodic Theory Dynam. Systems, 26 (2006), 1061. doi: 10.1017/S0143385706000204.
[13]	H. Furstenberg, "Recurrence in Ergodic Theory and Combinatorial Number Theory,", M. B. Porter Lectures, (1981).
[14]	Benjamin Green and Terence Tao, Linear equations in primes,, Ann. of Math., 171 (2010), 1753. 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. 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. 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. 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.
[19]	Bernard Host and Bryna Kra, Uniformity seminorms on $l^\infty$ and applications,, J. Anal. Math., 108 (2009), 219. 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. 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).
[22]	Jean-Pierre Kahane, "Some Random Series of Functions,", Second edition, 5 (1985).
[23]	A. Leibman, Polynomial mappings of groups,, Israel J. Math., 129 (2002), 29. doi: 10.1007/BF02773152.
[24]	A. Leibman, Convergence of multiple ergodic averages along polynomials of several variables,, Israel J. Math., 146 (2005), 303. 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. 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. 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. 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.
[29]	Elon Lindenstrauss, Pointwise theorems for amenable groups,, Invent. Math., 146 (2001), 259. doi: 10.1007/s002220100162.
[30]	A. I. Mal'cev, On a class of homogeneous spaces,, Izvestiya Akad. Nauk. SSSR. Ser. Mat., 13 (1949), 9.
[31]	E. Arthur Robinson, Jr., On uniform convergence in the Wiener-Wintner theorem,, J. London Math. Soc., 49 (1994), 493. 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. doi: 10.1017/S0143385700006921.
[33]	Terence Tao, "Higher Order Fourier Analysis,", Graduate Studies in Mathematics, 142 (2012).
[34]	Peter Walters, "An Introduction to Ergodic Theory,", Graduate Texts in Mathematics, 79 (1982).
[35]	Norbert Wiener and Aurel Wintner, Harmonic analysis and ergodic theory,, Amer. J. Math., 63 (1941), 415.
[36]	Pavel Zorin-Kranich, A nilpotent IP polynomial multiple recurrence theorem,, preprint, (2012).

show all references

References:

[1]	Robert A. Adams and John J. F. Fournier, "Sobolev Spaces,", Second edition, 140 (2003).
[2]	Idris Assani and Kimberly Presser, Pointwise characteristic factors for the multiterm return times theorem,, Ergodic Theory Dynam. Systems, 32 (2012), 341.
[3]	Idris Assani, "Wiener Wintner Ergodic Theorems,", World Scientific Publishing Co., (2003).
[4]	Idris Assani, Pointwise convergence of ergodic averages along cubes,, J. Anal. Math., 110 (2010), 241. 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. doi: 10.1007/s11511-007-0015-y.
[6]	J. Bourgain, Double recurrence and almost sure convergence,, J. Reine Angew. Math., 404 (1990), 140. doi: 10.1515/crll.1990.404.140.
[7]	S. Butkevich, "Convergence of Averages in Ergodic Theory,", Ph.D. thesis, (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. doi: 10.1112/plms/pdq037.
[9]	Qing Chu, Convergence of weighted polynomial multiple ergodic averages,, Proc. Amer. Math. Soc., 137 (2009), 1363. 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. doi: 10.1007/BF01673506.
[11]	Tanja Eisner and Terence Tao, Large values of the Gowers-Host-Kra seminorms,, J. Anal. Math., 117 (2012), 133. doi: 10.1007/s11854-012-0018-2.
[12]	Nikos Frantzikinakis, Uniformity in the polynomial Wiener-Wintner theorem,, Ergodic Theory Dynam. Systems, 26 (2006), 1061. doi: 10.1017/S0143385706000204.
[13]	H. Furstenberg, "Recurrence in Ergodic Theory and Combinatorial Number Theory,", M. B. Porter Lectures, (1981).
[14]	Benjamin Green and Terence Tao, Linear equations in primes,, Ann. of Math., 171 (2010), 1753. 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. 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. 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. 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.
[19]	Bernard Host and Bryna Kra, Uniformity seminorms on $l^\infty$ and applications,, J. Anal. Math., 108 (2009), 219. 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. 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).
[22]	Jean-Pierre Kahane, "Some Random Series of Functions,", Second edition, 5 (1985).
[23]	A. Leibman, Polynomial mappings of groups,, Israel J. Math., 129 (2002), 29. doi: 10.1007/BF02773152.
[24]	A. Leibman, Convergence of multiple ergodic averages along polynomials of several variables,, Israel J. Math., 146 (2005), 303. 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. 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. 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. 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.
[29]	Elon Lindenstrauss, Pointwise theorems for amenable groups,, Invent. Math., 146 (2001), 259. doi: 10.1007/s002220100162.
[30]	A. I. Mal'cev, On a class of homogeneous spaces,, Izvestiya Akad. Nauk. SSSR. Ser. Mat., 13 (1949), 9.
[31]	E. Arthur Robinson, Jr., On uniform convergence in the Wiener-Wintner theorem,, J. London Math. Soc., 49 (1994), 493. 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. doi: 10.1017/S0143385700006921.
[33]	Terence Tao, "Higher Order Fourier Analysis,", Graduate Studies in Mathematics, 142 (2012).
[34]	Peter Walters, "An Introduction to Ergodic Theory,", Graduate Texts in Mathematics, 79 (1982).
[35]	Norbert Wiener and Aurel Wintner, Harmonic analysis and ergodic theory,, Amer. J. Math., 63 (1941), 415.
[36]	Pavel Zorin-Kranich, A nilpotent IP polynomial multiple recurrence theorem,, preprint, (2012).

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