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On the Bogolyubov–Ruzsa lemma
, 2012
"... Our main result is that if A is a finite subset of an abelian group with jACAj 6 KjAj, then 2A 2A contains an O.logO.1 / 2K/dimensional coset progression M of size at least exp.O.logO.1 / 2K//jAj. ..."
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Our main result is that if A is a finite subset of an abelian group with jACAj 6 KjAj, then 2A 2A contains an O.logO.1 / 2K/dimensional coset progression M of size at least exp.O.logO.1 / 2K//jAj.
Approximate groups and their applications: work of Bourgain
 Gamburd, Helfgott, and Sarnak. Current Events Bulletin, AMS
"... Abstract. This is a survey of several exciting recent results in which techniques originating in the area known as additive combinatorics have been applied to give results in other areas, such as group theory, number theory and theoretical computer science. We begin with a discussion of the notion ..."
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Abstract. This is a survey of several exciting recent results in which techniques originating in the area known as additive combinatorics have been applied to give results in other areas, such as group theory, number theory and theoretical computer science. We begin with a discussion of the notion of an approximate group and also that of an approximate field, describing key results of FrĕımanRuzsa, BourgainKatzTao, Helfgott and others in which the structure of such objects is elucidated. We then move on to the applications. In particular we will look at the work of Bourgain and Gamburd on expansion properties of Cayley graphs on SL2(Fp) and at its application in the work of Bourgain, Gamburd and Sarnak on nonlinear sieving problems. 1.
Proposition 1.5, Remark 1.6.)
"... Let G be a group with a metric d invariant under left and right translations. A (K, r)approximate subgroup is a subset X of G containing 1, such that the product set XX is covered by at most K translates of XBr, where Br is the ball of radius r around 1. This notion is introduced by Terry Tao’s in ..."
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Let G be a group with a metric d invariant under left and right translations. A (K, r)approximate subgroup is a subset X of G containing 1, such that the product set XX is covered by at most K translates of XBr, where Br is the ball of radius r around 1. This notion is introduced by Terry Tao’s in his blog entry [3] 1 In this note, we generalize to this setting some of the results of [6].The modeltheoretic presentation uses additional assumptions; there may well be other routes. Here is the result under the strongest variant of the assumptions, a ’polynomial decay’ condition on the entropy of X as a function of the scale. (See also Proposition 1.4,