On a generalization of Szemerédi's theorem

• Autores: Ilya D. Shkredov
• Localización: Proceedings of the London Mathematical Society, ISSN 0024-6115, Vol. 93, Nº 3, 2006, págs. 723-760
• Idioma: inglés
• DOI: 10.1017/s0024611506015991
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• Resumen

  • Let $N$ be a natural number and $A \subseteq [1, \dots, N]^2$ be a set of cardinality at least $N^2 / (\log \log N)^c$, where $c > 0$ 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.