MATRIX PROGRESSIONS IN MULTIDIMENSIONAL SETS OF INTEGERS. Issue 1 (13th August 2014) - 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, …, N}^2 lacking four corners in a square has size at most CN^2(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].