MATRIX PROGRESSIONS IN MULTIDIMENSIONAL SETS OF INTEGERS. Issue 1 (January 2015) - Prendiville, Sean
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].