On a generalization of Szemerédi’s theorem. (English. Russian original) Zbl 1219.11017

Shkredov, I. D.

Dokl. Math. 72, No. 3, 899-902 (2005); translation from Dokl. Akad. Nauk, Ross. Akad. Nauk 405, No. 3, 315-319 (2005).

Summary: Let \(A \subseteq [1,..,N]^2\) 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 progression.

MSC:

11B25	Arithmetic progressions
11B75	Other combinatorial number theory
37A45	Relations of ergodic theory with number theory and harmonic analysis (MSC2010)

Citations:

Zbl 0015.15203; Zbl 0907.11005; Zbl 0303.10056; JFM 62.1126.01