Noncommutative sets of small doubling

Abstract

A corollary of Kneser's theorem, one sees that any finite nonempty subset A of an abelian group G = (G, +) with |A + A| ≤ (2 − ε)|A| can be covered by at most 2 ε −1 translates of a finite group H of cardinality at most (2 − ε)|A|. Using some arguments of Hamidoune, we establish an analogue in the noncommutative setting. Namely, if A is a finite non-empty subset of a nonabelian group G = (G, ·) such that |A · A| ≤ (2 − ε)|A|, then A is either contained in a right-coset of a finite group H of cardinality at most 2 ε |A|, or can be covered by at most 2 ε − 1 right-cosets of a finite group H of cardinality at most |A|. We also note some connections with some recent work of Sanders and of Petridis.

Additive combinatorics methods in associative algebras

“…The notion of connectivity for a subset S of a group G was developed by Hamidoune in [7]. As suggested by Tao in [17], it is interesting to generalize Hamidoune's definition by introducing an additional parameter λ. The purpose of this paragraph is to adapt this notion of connectivity to our algebra context.…”

“…Then by Lemma 2.1, V is a subalgebra containing k. In general, a space of small doubling k A is not a subalgebra and neither a left nor right H-module for a subalgebra k ⊆ H ⊆ A. The following theorem, which is a linear version of Theorem 1.2 in [17], permits to study the spaces of small doubling in an algebra A satisfying H s .…”

“…Nevertheless, there exist in this case numerous weaker results. Let us mention among them those of Diderrich [2], Olson [15] and Tao [17], [18] we shall evoke in more details in Section 4. Analogous estimates exist in the context of fields and division rings. As far as we are aware, this kind of generalizations was considered for the first time in [6] and [9].…”

Small Doubling in Groups

“…He showed (among other things) that if σ[A] < 3/2, then H := A·A −1 is a finite group of order |A·A| = σ[A]|A|, and that A ⊆ xH = Hx for some x. In a similar vein, an argument of Hamidoune [40,79] shows that if σ[A] < 2 − ε for some ε > 0, then there exists a finite group H of order |H| 2 ε |A|, such that A can be covered by at most 2 ε − 1 right-cosets Hx of H. See also [67] for a different proof of a related result. Very recently, a more complete classification of the sets A with σ[A] < 2 was achieved in [14].…”

Enumeration of three term arithmetic progressions in fixed density sets

“…To get the result to also hold for non-abelian groups they had to avoid using Fourier analysis. Other results in additive combinatorics for non-abelian groups are the results by Bergelson and Hindman [5], Gowers [20], Tao [76,77,78], Sanders [59] and Solymosi [69]. The results have been established long after the commutative counterparts, and often with much weaker bounds.…”