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Proceedings of the London Mathematical Society 2006 93(3):723-760; doi:10.1017/S0024611506015991
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© London Mathematical Society

On a Generalization of Szemerédi's Theorem

I. D. Shkredov

Department of Mechanics and Mathematics, Moscow State University Leninskie Gory, Moscow, 119992, Russia ishkredov{at}rambler.ru, ishkredo{at}mech.math.msu.su

Received 15 May 2005. Revision received 10 January 2006.

Let N be a natural number and A sub [1, ..., N]2 be a set of cardinality at least Formula 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.

This work was supported by the program ‘Leading Scientific Schools’ (project no. 136.2003.1), and by RFFI grant no. 02-01-00912 and INTAS (grant no. 03-51-5-70).


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