论文信息 - On a Generalization of Szemerédi's Theorem

On a Generalization of Szemerédi's Theorem

Let N be a natural number and A ⊂ [1, …, N]2 be a set of cardinality at least N2/(loglog⁡N)c is an absolute constant. We prove that A contains a triple {(k, m), (k+d, m), (k, m+d)}, where d > 0. This theorem is a two‐dimensional generalization of Szemerédi's theorem on arithmetic progressions. 2000 Mathematics Subject Classification 35J25, 37A15.

I. Shkredov

[1] R. Salem,et al. On Sets of Integers Which Contain No Three Terms in Arithmetical Progression. , 1942, Proceedings of the National Academy of Sciences of the United States of America.

[2] K. F. Roth. On Certain Sets of Integers , 1953 .

[3] E. Szemerédi. On sets of integers containing k elements in arithmetic progression , 1975 .

[4] Fan Chung Graham,et al. Quasi-random graphs , 1988, Comb..

[5] Peter J. Cameron,et al. Some sequences of integers , 1989, Discret. Math..

[6] Jean Bourgain,et al. On Triples in Arithmetic Progression , 1999 .

[7] W. T. Gowers,et al. A new proof of Szemerédi's theorem , 2001 .

[8] J. Solymosi. Note on a Generalization of Roth’s Theorem , 2003 .

[9] On a question of Gowers , 2003 .

[10] B. Green. A Szemerédi-type regularity lemma in abelian groups, with applications , 2003, math/0310476.

[11] On a question of Gowers concerning isosceles right-angle triangles , 2003 .

[13] T. Tao,et al. The primes contain arbitrarily long arithmetic progressions , 2004, math/0404188.

[14] Ilya D. Shkredov,et al. On a problem of Gowers , 2006 .

[15] G. Freiman. Foundations of a Structural Theory of Set Addition , 2007 .