MATRIX PROGRESSIONS IN MULTIDIMENSIONAL SETS OF INTEGERS. Issue 1 (13th August 2014)

• Record Type: Journal Article
• Authors: Prendiville, Sean
• Is Part Of: Mathematika. Volume 61:Issue 1(2015)
• Page Start: 14
• Page End: 48
• Publication Date: 2014-08-13
• Subjects: 11B30 (primary); Mathematics -- Periodicals; 510.5
• DOI: 10.1112/S0025579314000163
• Languages: English
• ISSNs: 0025-5793

Abstract

Abstract: We obtain density estimates for subsets of the $n$ -dimensional integer lattice lacking four-term matrix progressions. As a consequence, we show that a subset of the grid $\{1, 2, \dots, N\}^{2}$ lacking four corners in a square has size at most $\mathit{CN}^{2}(\log \log N)^{-c}$ . Our proofs involve the density increment method of Roth [ J. London Math. Soc. 28 (1953), 104–109] and Gowers [ Geom. Funct. Anal. 11 (3) (2001), 465–588], together with the $U^{3}$ -inverse theorem of Green and Tao [ Proc. Edinb. Math. Soc. (2)51 (1) (2008), 73–153].