MATRIX PROGRESSIONS IN MULTIDIMENSIONAL SETS OF INTEGERS. Issue 1 (January 2015)

• Record Type: Journal Article
• Authors: Prendiville, Sean
• Is Part Of: Mathematika. Volume 61:Issue 1(2015)
• Page Start: 14
• Page End: 48
• Publication Date: 2015-01
• Subjects: Mathematics -- Periodicals; 510.5
• DOI: 10.1112/S0025579314000163
• Languages: English
• ISSNs: 0025-5793

Abstract

<abstract> <title>Abstract</title> <p>We obtain density estimates for subsets of the <inline-formula><alternatives><inline-graphic xlink:href="ark:/27927/pgj46p9cp7" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><tex-math><![CDATA[$n$]]></tex-math></alternatives></inline-formula>-dimensional integer lattice lacking four-term matrix progressions. As a consequence, we show that a subset of the grid <inline-formula><alternatives><inline-graphic xlink:href="ark:/27927/pgj46p9bxc" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><tex-math><![CDATA[$\{1, 2, \dots, N\}^{2}$]]></tex-math></alternatives></inline-formula> lacking four corners in a square has size at most <inline-formula><alternatives><inline-graphic xlink:href="ark:/27927/pgj46p991f" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><tex-math><![CDATA[$\mathit{CN}^{2}(\log \log N)^{-c}$]]></tex-math></alternatives></inline-formula>. Our proofs involve the density increment method of Roth [<italic>J. London Math. Soc.</italic><bold>28</bold> (1953), 104–109] and Gowers [<italic>Geom. Funct. Anal.</italic><bold>11</bold>(3) (2001), 465–588], together with the <inline-formula><alternatives><inline-graphic xlink:href="ark:/27927/pgj46p9b71" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><tex-math><![CDATA[$U^{3}$]]></tex-math></alternatives></inline-formula>-inverse theorem of Green and Tao [<italic>Proc. Edinb. Math. Soc.</italic> (2) <bold>51</bold>(1) (2008), 73–153].</p> </abstract>