# 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, 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.  Google Scholar [2] Idris Assani and Kimberly Presser, Pointwise characteristic factors for the multiterm return times theorem, Ergodic Theory Dynam. Systems, 32 (2012), 341-360.  Google Scholar [3] Idris Assani, "Wiener Wintner Ergodic Theorems," World Scientific Publishing Co., Inc., River Edge, NJ, 2003.  Google Scholar [4] Idris Assani, Pointwise convergence of ergodic averages along cubes, J. Anal. Math., 110 (2010), 241-269. doi: 10.1007/s11854-010-0006-3.  Google Scholar [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.  Google Scholar [6] J. Bourgain, Double recurrence and almost sure convergence, J. Reine Angew. Math., 404 (1990), 140-161. doi: 10.1515/crll.1990.404.140.  Google Scholar [7] S. Butkevich, "Convergence of Averages in Ergodic Theory," Ph.D. thesis, Ohio State University, 2000.  Google Scholar [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.  Google Scholar [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.  Google Scholar [10] Andrés del Junco and Joseph Rosenblatt, Counterexamples in ergodic theory and number theory, Math. Ann., 245 (1979), 185-197. doi: 10.1007/BF01673506.  Google Scholar [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.  Google Scholar [12] Nikos Frantzikinakis, Uniformity in the polynomial Wiener-Wintner theorem, Ergodic Theory Dynam. Systems, 26 (2006), 1061-1071. doi: 10.1017/S0143385706000204.  Google Scholar [13] H. Furstenberg, "Recurrence in Ergodic Theory and Combinatorial Number Theory," M. B. Porter Lectures, Princeton University Press, Princeton, N.J., 1981.  Google Scholar [14] Benjamin Green and Terence Tao, Linear equations in primes, Ann. of Math., 171 (2010), 1753-1850. doi: 10.4007/annals.2010.171.1753.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [18] Bernard Host and Bryna Kra, Analysis of two step nilsequences, Ann. Inst. Fourier (Grenoble), 58 (2008), 1407-1453.  Google Scholar [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.  Google Scholar [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.  Google Scholar [21] Bernard Host, Bryna Kra and Alejandro Maass, Complexity of nilsystems and systems lacking nilfactors, preprint, (2012), arXiv:1203.3778. Google Scholar [22] Jean-Pierre Kahane, "Some Random Series of Functions," Second edition, Cambridge Studies in Advanced Mathematics, 5, Cambridge University Press, Cambridge, 1985.  Google Scholar [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.  Google Scholar [24] A. Leibman, Convergence of multiple ergodic averages along polynomials of several variables, Israel J. Math., 146 (2005), 303-315. doi: 10.1007/BF02773538.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [28] E. Lesigne, Spectre quasi-discret et théorème ergodique de Wiener-Wintner pour les polynômes, Ergodic Theory Dynam. Systems, 13 (1993), 767-784.  Google Scholar [29] Elon Lindenstrauss, Pointwise theorems for amenable groups, Invent. Math., 146 (2001), 259-295. doi: 10.1007/s002220100162.  Google Scholar [30] A. I. Mal'cev, On a class of homogeneous spaces, Izvestiya Akad. Nauk. SSSR. Ser. Mat., 13 (1949), 9-32.  Google Scholar [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.  Google Scholar [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.  Google Scholar [33] Terence Tao, "Higher Order Fourier Analysis," Graduate Studies in Mathematics, 142, American Mathematical Society, Providence, RI, 2012.  Google Scholar [34] Peter Walters, "An Introduction to Ergodic Theory," Graduate Texts in Mathematics, 79, Springer-Verlag, New York-Berlin, 1982.  Google Scholar [35] Norbert Wiener and Aurel Wintner, Harmonic analysis and ergodic theory, Amer. J. Math., 63 (1941), 415-426.  Google Scholar [36] Pavel Zorin-Kranich, A nilpotent IP polynomial multiple recurrence theorem, preprint, 2012, arXiv:1206.0287. Google Scholar

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.  Google Scholar [2] Idris Assani and Kimberly Presser, Pointwise characteristic factors for the multiterm return times theorem, Ergodic Theory Dynam. Systems, 32 (2012), 341-360.  Google Scholar [3] Idris Assani, "Wiener Wintner Ergodic Theorems," World Scientific Publishing Co., Inc., River Edge, NJ, 2003.  Google Scholar [4] Idris Assani, Pointwise convergence of ergodic averages along cubes, J. Anal. Math., 110 (2010), 241-269. doi: 10.1007/s11854-010-0006-3.  Google Scholar [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.  Google Scholar [6] J. Bourgain, Double recurrence and almost sure convergence, J. Reine Angew. Math., 404 (1990), 140-161. doi: 10.1515/crll.1990.404.140.  Google Scholar [7] S. Butkevich, "Convergence of Averages in Ergodic Theory," Ph.D. thesis, Ohio State University, 2000.  Google Scholar [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.  Google Scholar [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.  Google Scholar [10] Andrés del Junco and Joseph Rosenblatt, Counterexamples in ergodic theory and number theory, Math. Ann., 245 (1979), 185-197. doi: 10.1007/BF01673506.  Google Scholar [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.  Google Scholar [12] Nikos Frantzikinakis, Uniformity in the polynomial Wiener-Wintner theorem, Ergodic Theory Dynam. Systems, 26 (2006), 1061-1071. doi: 10.1017/S0143385706000204.  Google Scholar [13] H. Furstenberg, "Recurrence in Ergodic Theory and Combinatorial Number Theory," M. B. Porter Lectures, Princeton University Press, Princeton, N.J., 1981.  Google Scholar [14] Benjamin Green and Terence Tao, Linear equations in primes, Ann. of Math., 171 (2010), 1753-1850. doi: 10.4007/annals.2010.171.1753.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [18] Bernard Host and Bryna Kra, Analysis of two step nilsequences, Ann. Inst. Fourier (Grenoble), 58 (2008), 1407-1453.  Google Scholar [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.  Google Scholar [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.  Google Scholar [21] Bernard Host, Bryna Kra and Alejandro Maass, Complexity of nilsystems and systems lacking nilfactors, preprint, (2012), arXiv:1203.3778. Google Scholar [22] Jean-Pierre Kahane, "Some Random Series of Functions," Second edition, Cambridge Studies in Advanced Mathematics, 5, Cambridge University Press, Cambridge, 1985.  Google Scholar [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.  Google Scholar [24] A. Leibman, Convergence of multiple ergodic averages along polynomials of several variables, Israel J. Math., 146 (2005), 303-315. doi: 10.1007/BF02773538.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [28] E. Lesigne, Spectre quasi-discret et théorème ergodique de Wiener-Wintner pour les polynômes, Ergodic Theory Dynam. Systems, 13 (1993), 767-784.  Google Scholar [29] Elon Lindenstrauss, Pointwise theorems for amenable groups, Invent. Math., 146 (2001), 259-295. doi: 10.1007/s002220100162.  Google Scholar [30] A. I. Mal'cev, On a class of homogeneous spaces, Izvestiya Akad. Nauk. SSSR. Ser. Mat., 13 (1949), 9-32.  Google Scholar [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.  Google Scholar [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.  Google Scholar [33] Terence Tao, "Higher Order Fourier Analysis," Graduate Studies in Mathematics, 142, American Mathematical Society, Providence, RI, 2012.  Google Scholar [34] Peter Walters, "An Introduction to Ergodic Theory," Graduate Texts in Mathematics, 79, Springer-Verlag, New York-Berlin, 1982.  Google Scholar [35] Norbert Wiener and Aurel Wintner, Harmonic analysis and ergodic theory, Amer. J. Math., 63 (1941), 415-426.  Google Scholar [36] Pavel Zorin-Kranich, A nilpotent IP polynomial multiple recurrence theorem, preprint, 2012, arXiv:1206.0287. Google Scholar
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