A Szemerédi-type regularity lemma in abelian groups, with applications

Abstract

Abstract. Szemerédi's regularity lemma is an important tool in graph theory which has applications throughout combinatorics.In this paper we prove an analogue of Szemerédi's regularity lemma in the context of abelian groups and use it to derive some results in additive number theory.One is a structure theorm for sets which are almost sum-free. If A ⊆ {1, . . . , N } has δN 2 triples (a 1 , a 2 , a 3 ) for which a 1 + a 2 = a 3 then A = B ∪ C, where B is sum-free and |C| = δ ′ N , and δ ′ → 0 as δ → 0Another answers a question of Bergelson, Host and Kra. If α, ǫ > 0, if N > N 0 (α, ǫ) and if A ⊆ {1, . . . , N } has size αN , then there is some d = 0 such that A contains at least (α 3 − ǫ)N three-term arithmetic progressions with common difference d.

The symmetry preserving removal lemma

“…The symmetry preserving removal lemma yields that if ab = c has o(|T | 2 ) solutions in S, then one can remove o(|T |) elements from S such that in the new set there is no solution to ab = c. This was first proved by Ben Green [2] for Abelian groups and generalized to groups by Král, Serra and Vena [4].…”

Odd circuits in dense binary matroids

“…The following result of Green [3] guarantees a regular subspace of bounded codimension. Here W (t) denotes an exponential tower of 2's of height ⌈t⌉.…”

“…We obtain the theorem as an easy application of Green's regularity lemma for finite abelian groups [3], which we review in Section 2.…”

Counting configuration–free sets in groups

“…Therefore, we can use Proposition 6.5 to show concentration around the mean in this interval. In particular, we can apply Theorem 5.5 to obtain the 0-statement in the region defined by (11) and (20) as there is an asymptotic gap between the two by (19), and additionally (21) holds.…”

“…The idea behind them can be traced back to the proof of Roth's Theorem by Ruzsa and Szemerédi [30] and was first formulated by Green [19] for a linear equation in an abelian group by using methods of Fourier analysis. The picture was complemented independently by Shapira [36] and by Král', Serra and Vena [23] by proving a removal lemma for linear systems in the integers.…”