The Inverse Conjecture for the Gowers Norm over Finite Fields in Low Characteristic
Tóm tắt
Từ khóa
Tài liệu tham khảo
Alon, N., Kaufman, T., Krivelevich, M., Litsyn, S., Ron, D.: Testing flow-degree polynomials over GF(2). In: Arora, S. et al. (eds.) Approximation, Randomization, and Combinatorial Optimization, pp. 188–199. Spinger, Berlin (2003). Also: Testing Reed-Muller codes. IEEE Trans. Inform. Theory 51(11), 4032–4039 (2005)
Austin T.: On the norm convergence of nonconventional ergodic averages. Ergodic Theory Dynam. Systems 30(2), 321–338 (2010)
Bergelson V., Tao T., Ziegler T.: An inverse theorem for the uniformity seminorms associated with the action of $${\mathbb{F}_p^\infty}$$ . Geom. Funct. Anal. 19(6), 1539–1596 (2010)
Bergelson V., Leibman A., McCutcheon R.: Polynomial Szemerédi theorems for countable modules over integral domains and finite fields. J. Anal. Math. 95, 243–296 (2005)
Camarena, O., Szegedy, B.: Nilspaces, nilmanifolds and their morphisms. Preprint. arXiv:1009.3825 (2010)
Frantzikinakis N., Host B., Kra B.: Multiple recurrence and convergence for sequences related to the prime numbers. J. Reine Angew. Math. 611, 131–144 (2007)
Furstenberg H.: Ergodic behavior of diagonal measures and a theorem of Szemerédi on arithmetic progressions. J. Anal. Math. 31, 204–256 (1977)
Furstenberg H., Katznelson Y.: A density version of the Hales-Jewett theorem. J. Anal. Math. 57, 64–119 (1991)
Gödel K.: The consistency of the axiom of choice and of the generalized continuumhypothesis. Proc. Natl. Acad. Sci. USA 24(12), 556–557 (1938)
Gowers T.: A new proof of Szemerédi’s theorem for arithmetic progressions of length four. Geom. Funct. Anal. 8(3), 529–551 (1998)
Gowers T.: Decompositions, approximate structure, transference, and the Hahn-Banach theorem. Bull. Lond. Math. Soc. 42(4), 573–606 (2010)
Gowers T., Wolf J.: The true complexity of a system of linear equations. Proc. Lond. Math. Soc. (3) 100(1), 155–176 (2010)
Gowers, T., Wolf, J.: Linear forms and quadratic uniformity for functions on $${\mathbb{F}_p^n}$$ . Preprint.
Gowers T., Wolf J.: Linear forms and higher-degree uniformity for functions on $${\mathbb{F}_p^n}$$ . Geom. Funct. Anal. 21(1), 36–69 (2011)
Green B., Tao T.: The primes contain arbitrarily long arithmetic progressions. Ann. of Math. 167(2), 481–547 (2008)
Green B., Tao T.: An inverse theorem for the Gowers U 3(G) norm. Proc. Edinburgh Math. Soc. (2) 51(1), 73–153 (2008)
Green B., Tao T.: New bounds for Szemerédi’s Theorem, I: progressions of length 4 in finite field geometries. Proc. Lond. Math. Soc. (3) 98(2), 365–392 (2009)
Green, B., Tao, T.: The quantitative behaviour of polynomial orbits on nilmanifolds. Ann. of Math. (2) (to appear)
Green B., Tao T.: The distribution of polynomials over finite fields, with applications to the Gowers norms. Contrib. Discrete Math. 4(2), 1–36 (2009)
Green, B., Tao, T.: An arithmetic regularity lemma, an associated counting lemma, and applications. Preprint
Green, B., Tao, T., Ziegler, T.: An inverse theorem for the Gowers U s+1[N] norm. Preprint
Hall P.: A contribution to the theory of groups of prime-power order. Proc. London Math. Soc. (2) 36(1), 29–95 (1934)
Host B., Kra B.: Nonconventional ergodic averages and nilmanifolds. Ann. of Math. (2) 161(1), 397–488 (2005)
Host B., Kra B.: Parallelepipeds, nilpotent groups and Gowers norms. Bull. Soc. Math. France 136(3), 405–437 (2008)
Kaufman, T., Lovett, S.: Worst case to average case reductions for polynomials. In: Proceedings of the 49th Annual IEEE Symposium on Foundations of Computer Science, pp. 166–175. IEEE Computer Society, Los Alamitos, CA (2008)
Lazard M.: Sur certaines suites d’éléments dans les groupes libres et leurs extensions. C. R. Acad. Sci. Paris 236, 36–38 (1953)
Loeb P.A.: Conversion from nonstandard to standard measure spaces and applications in probability theory. Trans. Amer. Math. Soc. 211, 113–122 (1975)
Lovett, S., Meshulam, R., Samorodnitsky, A.: Inverse conjecture for the Gowers norm is false. In: Dwork, C. (ed.) Proceedings of the 40th Annual ACM Symposium on Theory of Computing held in Victoria, BC, May 17-20, 2008, pp. 547–556. ACM, New York (2008)
Samorodnitsky, A.: Low-degree tests at large distances. In: Feige, U. (ed.) STOC’07–Proceedings of the 39th Annual ACM Symposium on Theory of Computing, San Diego, California, USA, June 11-13, 2007, pp. 506–515. ACM, New York (2007)
Sudan M., Trevisan L., Vadhan S.: Pseudorandom generators without the XOR lemma. J. Comput. System Sci. 62(2), 236–266 (2001)
Szegedy, B.: Gowers norms, regularization and limits of functions on abelian groups. Preprint. Avaible at arXiv:1010.6211 (2010)
Szegedy, B.: Structure of finite nilspaces and inverse theorems for the Gowers norms in bounded exponent groups. Preprint. Avaible at arXiv:1011.1057 (2010)
Tao T.: A quantitative ergodic theory proof of Szemerédi’s theorem. Electron. J. Combin. 13(1), #R99 (2006)
Tao, T.: Structure and randomness in combinatorics. In: FOCS’07: Proceedings of the 48th Annual Symposium on Foundations of Computer Science, pp. 3–18. IEEE Computer Society, Los Alamitos (2007)
Tao, T.: Poincaré’s Legacies, Vol I., American Mathematical Society, Providence (2009)
Tao T., Ziegler T.: The inverse conjecture for the Gowers norm over finite fields via the correspondence principle. Anal. PDE 3(1), 1–20 (2010)
Towsner, H.: A correspondence principle for the Gowers norms. Preprint. Avaible at arXiv:0905.0493 (2009)