MATRIX PROGRESSIONS IN MULTIDIMENSIONAL SETS OF INTEGERS. Issue 1 (January 2015) - Prendiville, Sean
Abstract We obtain density estimates for subsets of the <alternatives><inline-graphic xlink:href="ark:/27927/pgj46p9cp7" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /> <![CDATA[$n$]]> </alternatives>-dimensional integer lattice lacking four-term matrix progressions. As a consequence, we show that a subset of the grid <alternatives><inline-graphic xlink:href="ark:/27927/pgj46p9bxc" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /> <![CDATA[$\{1, 2, \dots, N\}^{2}$]]> </alternatives> lacking four corners in a square has size at most <alternatives><inline-graphic xlink:href="ark:/27927/pgj46p991f" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /> <![CDATA[$\mathit{CN}^{2}(\log \log N)^{-c}$]]> </alternatives>. 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 <alternatives><inline-graphic xlink:href="ark:/27927/pgj46p9b71" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /> <![CDATA[$U^{3}$]]> </alternatives>-inverse theorem of Green and Tao [Proc. Edinb. Math. Soc. (2) 51(1) (2008), 73–153].