TL;DR
This paper proves that in finite abelian groups with small sumsets, the difference set contains a large structured subset called a coset progression with controlled dimension and size.
Contribution
It establishes a bound on the structure of difference sets in abelian groups with small doubling, extending the understanding of additive combinatorics.
Findings
2A-2A contains a large coset progression
Dimension of the progression is polylogarithmic in K
Size of the progression is exponentially related to |A|
Abstract
Our main result is that if A is a finite subset of an abelian group with |A+A| < K|A|, then 2A-2A contains an O(log^{O(1)} K)-dimensional coset progression M of size at least exp(-O(log^{O(1)} K))|A|.
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