# American Institute of Mathematical Sciences

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

## 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

Received  May 2012 Revised  October 2012 Published  January 2013

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.
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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