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A few days ago, I received the sad news that Yahya Ould Hamidoune had recently died. Hamidoune worked in additive combinatorics, and had recently solved a question on noncommutative Freiman-Kneser theorems posed by myself on this blog last year. Namely, Hamidoune showed

Theorem 1 (Noncommutative Freiman-Kneser theorem for small doubling) Let {0 < \epsilon \leq 1}, and let {S \subset G} be a finite non-empty subset of a multiplicative group {G} such that {|A \cdot S| \leq (2-\epsilon) |S|} for some finite set {A} of cardinality {|A|} at least {|S|}, where {A \cdot S := \{ as: a \in A, s \in S \}} is the product set of {A} and {S}. Then there exists a finite subgroup {H} of {G} with cardinality {|H| \leq C(\epsilon) |S|}, such that {S} is covered by at most {C'(\epsilon)} right-cosets {H \cdot x} of {H}, where {c(\epsilon), C(\epsilon) > 0} depend only on {\epsilon}.

One can of course specialise here to the case {A=S}, and view this theorem as a classification of those sets {S} of doubling constant at most {2-\epsilon}.

In fact Hamidoune’s argument, which is completely elementary, gives the very nice explicit constants {C(\epsilon) := \frac{2}{\epsilon}} and {C'(\epsilon) := \frac{2}{\epsilon} - 1}, which are essentially optimal except for factors of {2} (as can be seen by considering an arithmetic progression in an additive group). This result was also independently established (in the {A=S} case) by Tom Sanders (unpublished) by a more Fourier-analytic method, in particular drawing on Sanders’ deep results on the Wiener algebra {A(G)} on arbitrary non-commutative groups {G}.

This type of result had previously been known when {2-\epsilon} was less than the golden ratio {\frac{1+\sqrt{5}}{2}}, as first observed by Freiman; see my previous blog post for more discussion.

Theorem 1 is not, strictly speaking, contained in Hamidoune’s paper, but can be extracted from his arguments, which share some similarity with the recent simple proof of the Ruzsa-Plünnecke inequality by Petridis (as discussed by Tim Gowers here), and this is what I would like to do below the fold. I also include (with permission) Sanders’ unpublished argument, which proceeds instead by Fourier-analytic methods. Read the rest of this entry »


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