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MATRIX PROGRESSIONS IN MULTIDIMENSIONAL SETS OF INTEGERS

Part of: Sequences and sets

Published online by Cambridge University Press:  13 August 2014

Sean Prendiville
Sean Prendiville*
Affiliation:
Department of Mathematics and Statistics, University of Reading, Whiteknights, PO Box 220, Reading, RG6 6AX, U.K. email s.m.prendiville@reading.ac.uk
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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].

MSC classification

Primary: 11B30: Arithmetic combinatorics; higher degree uniformity
Type
Research Article
Information
Mathematika , Volume 61 , Issue 1 , January 2015 , pp. 14 - 48
Copyright
Copyright © University College London 2014 

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