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On the structure of steps of three‐term arithmetic progressions in a dense set of integers
Author(s) -
Candela P.
Publication year - 2010
Publication title -
bulletin of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 2.396
H-Index - 48
eISSN - 1469-2120
pISSN - 0024-6093
DOI - 10.1112/blms/bdp074
Subject(s) - mathematics , arithmetic progression , term (time) , set (abstract data type) , arithmetic , combinatorics , decomposition , binary logarithm , quadratic equation , discrete mathematics , geometry , physics , quantum mechanics , computer science , programming language , ecology , biology
We use recent results in quadratic Fourier analysis to examine the additive structure of the set of steps (or common differences) of three‐term arithmetic progressions in a general subset of [ N ]={1, 2, …, N } of fixed positive density. In particular, combining the decomposition results of Gowers and Wolf with the recurrence results of Green and Tao, we show that if A ⊂ [ N ] has density α > 0, then, for some positive constant c = c (α), the set of steps of three‐term arithmetic progressions in A contains an arithmetic progression of length at least c (log log N ) c . This improves on the estimate of shape Ω α (log log log log log N ) that one can obtain by a straightforward application of Gowers' bounds for Szemerédi's theorem.