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MATRIX PROGRESSIONS IN MULTIDIMENSIONAL SETS OF INTEGERS
Author(s) -
Prendiville Sean
Publication year - 2015
Publication title -
mathematika
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.955
H-Index - 29
eISSN - 2041-7942
pISSN - 0025-5793
DOI - 10.1112/s0025579314000163
Subject(s) - mathematics , mathematical proof , inverse , integer (computer science) , combinatorics , matrix (chemical analysis) , square tiling , lattice (music) , grid , discrete mathematics , geometry , materials science , composite material , physics , computer science , acoustics , programming language
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 mostCN 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].