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New bounds for Szemerédi's theorem, I: progressions of length 4 in finite field geometries
Author(s) -
Green Ben,
Tao Terence
Publication year - 2009
Publication title -
proceedings of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.899
H-Index - 65
eISSN - 1460-244X
pISSN - 0024-6115
DOI - 10.1112/plms/pdn030
Subject(s) - cardinality (data modeling) , mathematics , combinatorics , abelian group , finite field , integer (computer science) , arithmetic progression , field (mathematics) , upper and lower bounds , constant (computer programming) , finite group , discrete mathematics , group (periodic table) , physics , pure mathematics , mathematical analysis , quantum mechanics , computer science , data mining , programming language
Let k ⩾ 3 be an integer, and let G be a finite abelian group with | G | = N , where ( N , ( k − 1)!) = 1. We write r k ( G ) for the largest cardinality | A | of a set A ⊆ G which does not contain k distinct elements in arithmetic progression. The famous theorem of Szemerédi essentially asserts that r k (ℤ/ N ℤ) = o k ( N ). It is known, in fact, that the estimate r k ( G ) = o k ( N ) holds for all G . There have been many papers concerning the issue of finding quantitative bounds for r k ( G ). A result of Bourgain states that r 3 ( G ) ≪ N (log log N /log N ) 1/2 for all G . In this paper we obtain a similar bound for r 4 ( G ) in the particular case G = F n , where F is a fixed finite field with char ( F ) ≠ 2, 3 (for example, F = 5 ). We prove that r 4 ( G ) ≪ F N (log N ) − c for some absolute constant c > 0. In future papers we will treat the general abelian groups G , eventually obtaining a comparable result for an arbitrary G .