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An inverse theorem for the Gowers $U^{s+1}[N]$-norm
1. | Centre for Mathematical Sciences, Wilberforce Road, Cambridge CB3 0WA |
2. | Department of Mathematics, UCLA, 405 Hilgard Ave, Los Angeles, CA 90095 |
3. | Department of Mathematics, Technion - Israel Institute of Technology, Haifa, 32000, Israel |
References:
[1] |
N. Alon, T. Kaufman, M. Krivelevich, S. Litsyn and D. Ron, Testing low-degree polynomials over GF(2), Approximation, Randomization, and Combinatorial Optimization, 2003, 188-199. Also: Testing Reed-Muller codes, IEEE Transactions on Information Theory, 51 (2005), 4032-4039.
doi: 10.1109/TIT.2005.856958. |
[2] |
A. Balog and E. Szemerédi, A statistical theorem of set addition, Combinatorica, 14 (1994), 263-268.
doi: 10.1007/BF01212974. |
[3] |
V. Bergelson, B. Host and B. Kra, Multiple recurrence and nilsequences, (with an appendix by I. Ruzsa), Invent. Math., 160 (2005), 261-303.
doi: 10.1007/s00222-004-0428-6. |
[4] |
V. Bergelson, T. Tao and T. Ziegler, An inverse theorem for uniformity seminorms associated with the action of $F_p^{\infty}$, Geom. Funct. Anal., 19 (2010), 1539-1596.
doi: 10.1007/s00039-010-0051-1. |
[5] |
J.-P. Conze and E. Lesigne, Sur un théorème ergodique pour des mesures diagonales, (French) [On an ergodic theorem for diagonal measures], C. R. Acad. Sci. Paris Sér. I Math., 306 (1988), 491-493. |
[6] |
N. Frantzikinakis, B. Host and B. Kra, Multiple recurrence and convergence for sequences related to the prime numbers, J. Reine Angew. Math., 611 (2007), 131-144.
doi: 10.1515/CRELLE.2007.076. |
[7] |
G. A. Freĭman, "Foundations of a Structural Theory of Set Addition," Translations of Mathematical Monographs, 37, AMS, Providence, RI, 1973. |
[8] |
H. Furstenberg, "Nonconventional Ergodic Averages," The legacy of John von Neumann (Hempstead, NY, 1988), 43-56, Proc. Sympos. Pure Math., 50, Amer. Math. Soc., Providence, RI, 1990. |
[9] |
H. Furstenberg and B. Weiss, "A mean ergodic theorem for $1/N\sum^N_{n=1}f (T^n x)g(T^ {n^ 2}x)$," Convergence in ergodic theory and probability (Columbus, OH, 1993), 193-227, Ohio State Univ. Math. Res. Inst. Publ., 5 de Gruyter, Berlin, 1996. |
[10] |
W. T. Gowers, A new proof of Szemerédi's theorem for progressions of length four, GAFA, 8 (1998), 529-551.
doi: 10.1007/s000390050065. |
[11] |
-, A new proof of Szemerédi's theorem, GAFA, 11 (2001), 465-588. |
[12] |
B. Green, "Generalising the Hardy-Littlewood Method for Primes," International Congress of Mathematicians. Vol. II, 373-399, Eur. Math. Soc., Zürich, 2006. |
[13] |
B. Green and T. Tao, The primes contain arbitrarily long arithmetic progressions, Annals of Math. (2), 167 (2008), 481-547.
doi: 10.4007/annals.2008.167.481. |
[14] |
-, An inverse theorem for the Gowers $U^3$-norm, with applications, Proc. Edinburgh Math. Soc., 51, 71-153. |
[15] |
-, Linear equations in primes, Ann. Math. (2), 171 (2010), 1753-1850. |
[16] |
-, The quantitative behaviour of polynomial orbits on nilmanifolds, to appear in Ann. Math. |
[17] |
-, The Möbius function is strongly orthogonal to nilsequences, to appear in Ann. Math. |
[18] |
-, An arithmetic regularity lemma, associated counting lemma, and applications, in "An Irregular Mind" (Szemerédi is 70), Bolyai Society Mathematical Studies, 21, 261-334. |
[19] |
B. Green, T. Tao and T. Ziegler, An inverse theorem for the Gowers $U^4[N]$-norm, Glasgow Math. J., 53 (2011), 1-50.
doi: 10.1017/S0017089510000546. |
[20] |
-, An inverse theorem for the Gowers $U^{s+1}[N]$ norm, preprint, arXiv:1009.3998. |
[21] |
I. J. Håland, Uniform distribution of generalized polynomials, J. Number Theory, 45 (1993), 327-366.
doi: 10.1006/jnth.1993.1082. |
[22] |
B. Host and B. Kra, Convergence of Conze-Lesigne averages, Erg. Th. Dyn. Sys., 21 (2001), 493-509. |
[23] |
-, Averaging along cubes, Modern Dynamical Systems and Applications, Cambridge University Press, Cambridge, (2004), 123-144. |
[24] |
-, Nonconventional ergodic averages and nilmanifolds, Ann. of Math. (2), 161 (2005), 397-488. |
[25] |
-, Uniformity seminorms on $l^\infty$ and applications, J. Anal. Math., 108 (2009), 219-276.
doi: 10.1007/s11854-009-0024-1. |
[26] |
A. Leibman, Pointwise convergence of ergodic averages of polynomial sequences of translations on a nilmanifold, Ergodic Theory and Dynamical Systems, 25 (2005), 201-213.
doi: 10.1017/S0143385704000215. |
[27] |
-, A canonical form and the distribution of values of generalized polynomials, to appear in Israel Journal of Mathematics. |
[28] |
E. Lesigne, Équations fonctionnelles, couplages de produits gauches et théorèmes ergodiques pour mesures diagonales, (French) [Functional equations, couplings of skew products, ergodic theorems for diagonal measures], Bull. Soc. Math. France, 121 (1993), 315-351. |
[29] |
I. Z. Ruzsa, Generalized arithmetical progressions and sumsets, Acta Math. Hungar., 65 (1994), 379-388.
doi: 10.1007/BF01876039. |
[30] |
A. Samorodnitsky, "Low-Degree Tests at Large Distances," STOC'07, Proceedings of the 39th Annual ACM Symposium on Theory of Computing, ACM, New York, (2007), 506-515. |
[31] |
B. Szegedy, Higher order Fourier analysis as an algebraic theory I, preprint, arXiv::0903.0897. |
[32] |
-, Higher order Fourier analysis as an algebraic theory II, preprint, arXiv::0911.1157. |
[33] |
-, Higher order Fourier analysis as an algebraic theory III, preprint, arXiv::1001.4282. |
[34] |
T. Tao and T. Ziegler, The inverse conjecture for the Gowers norms over finite fields via the correspondence principle, Anal. PDE, 3 (2010), 1-20.
doi: 10.2140/apde.2010.3.1. |
[35] |
T. Tao and V. Vu, "Additive Combinatorics," Cambridge Studies in Advanced Mathematics, 105, Cambridge University Press, Cambridge, 2006. |
[36] |
T. Ziegler, Universal characteristic factors and Furstenberg averages, J. Amer. Math. Soc., 20 (2007), 53-97.
doi: 10.1090/S0894-0347-06-00532-7. |
show all references
References:
[1] |
N. Alon, T. Kaufman, M. Krivelevich, S. Litsyn and D. Ron, Testing low-degree polynomials over GF(2), Approximation, Randomization, and Combinatorial Optimization, 2003, 188-199. Also: Testing Reed-Muller codes, IEEE Transactions on Information Theory, 51 (2005), 4032-4039.
doi: 10.1109/TIT.2005.856958. |
[2] |
A. Balog and E. Szemerédi, A statistical theorem of set addition, Combinatorica, 14 (1994), 263-268.
doi: 10.1007/BF01212974. |
[3] |
V. Bergelson, B. Host and B. Kra, Multiple recurrence and nilsequences, (with an appendix by I. Ruzsa), Invent. Math., 160 (2005), 261-303.
doi: 10.1007/s00222-004-0428-6. |
[4] |
V. Bergelson, T. Tao and T. Ziegler, An inverse theorem for uniformity seminorms associated with the action of $F_p^{\infty}$, Geom. Funct. Anal., 19 (2010), 1539-1596.
doi: 10.1007/s00039-010-0051-1. |
[5] |
J.-P. Conze and E. Lesigne, Sur un théorème ergodique pour des mesures diagonales, (French) [On an ergodic theorem for diagonal measures], C. R. Acad. Sci. Paris Sér. I Math., 306 (1988), 491-493. |
[6] |
N. Frantzikinakis, B. Host and B. Kra, Multiple recurrence and convergence for sequences related to the prime numbers, J. Reine Angew. Math., 611 (2007), 131-144.
doi: 10.1515/CRELLE.2007.076. |
[7] |
G. A. Freĭman, "Foundations of a Structural Theory of Set Addition," Translations of Mathematical Monographs, 37, AMS, Providence, RI, 1973. |
[8] |
H. Furstenberg, "Nonconventional Ergodic Averages," The legacy of John von Neumann (Hempstead, NY, 1988), 43-56, Proc. Sympos. Pure Math., 50, Amer. Math. Soc., Providence, RI, 1990. |
[9] |
H. Furstenberg and B. Weiss, "A mean ergodic theorem for $1/N\sum^N_{n=1}f (T^n x)g(T^ {n^ 2}x)$," Convergence in ergodic theory and probability (Columbus, OH, 1993), 193-227, Ohio State Univ. Math. Res. Inst. Publ., 5 de Gruyter, Berlin, 1996. |
[10] |
W. T. Gowers, A new proof of Szemerédi's theorem for progressions of length four, GAFA, 8 (1998), 529-551.
doi: 10.1007/s000390050065. |
[11] |
-, A new proof of Szemerédi's theorem, GAFA, 11 (2001), 465-588. |
[12] |
B. Green, "Generalising the Hardy-Littlewood Method for Primes," International Congress of Mathematicians. Vol. II, 373-399, Eur. Math. Soc., Zürich, 2006. |
[13] |
B. Green and T. Tao, The primes contain arbitrarily long arithmetic progressions, Annals of Math. (2), 167 (2008), 481-547.
doi: 10.4007/annals.2008.167.481. |
[14] |
-, An inverse theorem for the Gowers $U^3$-norm, with applications, Proc. Edinburgh Math. Soc., 51, 71-153. |
[15] |
-, Linear equations in primes, Ann. Math. (2), 171 (2010), 1753-1850. |
[16] |
-, The quantitative behaviour of polynomial orbits on nilmanifolds, to appear in Ann. Math. |
[17] |
-, The Möbius function is strongly orthogonal to nilsequences, to appear in Ann. Math. |
[18] |
-, An arithmetic regularity lemma, associated counting lemma, and applications, in "An Irregular Mind" (Szemerédi is 70), Bolyai Society Mathematical Studies, 21, 261-334. |
[19] |
B. Green, T. Tao and T. Ziegler, An inverse theorem for the Gowers $U^4[N]$-norm, Glasgow Math. J., 53 (2011), 1-50.
doi: 10.1017/S0017089510000546. |
[20] |
-, An inverse theorem for the Gowers $U^{s+1}[N]$ norm, preprint, arXiv:1009.3998. |
[21] |
I. J. Håland, Uniform distribution of generalized polynomials, J. Number Theory, 45 (1993), 327-366.
doi: 10.1006/jnth.1993.1082. |
[22] |
B. Host and B. Kra, Convergence of Conze-Lesigne averages, Erg. Th. Dyn. Sys., 21 (2001), 493-509. |
[23] |
-, Averaging along cubes, Modern Dynamical Systems and Applications, Cambridge University Press, Cambridge, (2004), 123-144. |
[24] |
-, Nonconventional ergodic averages and nilmanifolds, Ann. of Math. (2), 161 (2005), 397-488. |
[25] |
-, Uniformity seminorms on $l^\infty$ and applications, J. Anal. Math., 108 (2009), 219-276.
doi: 10.1007/s11854-009-0024-1. |
[26] |
A. Leibman, Pointwise convergence of ergodic averages of polynomial sequences of translations on a nilmanifold, Ergodic Theory and Dynamical Systems, 25 (2005), 201-213.
doi: 10.1017/S0143385704000215. |
[27] |
-, A canonical form and the distribution of values of generalized polynomials, to appear in Israel Journal of Mathematics. |
[28] |
E. Lesigne, Équations fonctionnelles, couplages de produits gauches et théorèmes ergodiques pour mesures diagonales, (French) [Functional equations, couplings of skew products, ergodic theorems for diagonal measures], Bull. Soc. Math. France, 121 (1993), 315-351. |
[29] |
I. Z. Ruzsa, Generalized arithmetical progressions and sumsets, Acta Math. Hungar., 65 (1994), 379-388.
doi: 10.1007/BF01876039. |
[30] |
A. Samorodnitsky, "Low-Degree Tests at Large Distances," STOC'07, Proceedings of the 39th Annual ACM Symposium on Theory of Computing, ACM, New York, (2007), 506-515. |
[31] |
B. Szegedy, Higher order Fourier analysis as an algebraic theory I, preprint, arXiv::0903.0897. |
[32] |
-, Higher order Fourier analysis as an algebraic theory II, preprint, arXiv::0911.1157. |
[33] |
-, Higher order Fourier analysis as an algebraic theory III, preprint, arXiv::1001.4282. |
[34] |
T. Tao and T. Ziegler, The inverse conjecture for the Gowers norms over finite fields via the correspondence principle, Anal. PDE, 3 (2010), 1-20.
doi: 10.2140/apde.2010.3.1. |
[35] |
T. Tao and V. Vu, "Additive Combinatorics," Cambridge Studies in Advanced Mathematics, 105, Cambridge University Press, Cambridge, 2006. |
[36] |
T. Ziegler, Universal characteristic factors and Furstenberg averages, J. Amer. Math. Soc., 20 (2007), 53-97.
doi: 10.1090/S0894-0347-06-00532-7. |
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