You are currently browsing the tag archive for the ‘tom sanders’ tag.

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 This result was also independently established (in the ${A=S}$ case) by Tom Sanders (unpublished) by a more Fourier-analytic method, in particular drawing on Sanders’ deep results on the Wiener algebra ${A(G)}$ on arbitrary non-commutative groups ${G}$.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 I also include (with permission) Sanders’ unpublished argument, which proceeds instead by Fourier-analytic methods.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 »

This is an adaptation of a talk I gave recently for a program at IPAM. In this talk, I gave a (very informal and non-rigorous) overview of Hrushovski’s use of model-theoretic techniques to establish new Freiman-type theorems in non-commutative groups, and some recent work in progress of Ben Green, Tom Sanders and myself to establish combinatorial proofs of some of Hrushovski’s results. Symbols on Eigenvectors from eigenvalues Anonymous on Analysis I Anonymous on Real stable polynomials and th… Anonymous on Real stable polynomials and th… BabaDaga on Homogenization of iterated sin… Hollis Williams on Homogenization of iterated sin… BabaDaga on Homogenization of iterated sin… Jaikrishnan Janardha… on The inverse function theorem f… DSK MATHS MATH TREAS… on Mathematics Seminars List Shameka on Displaying mathematics on the… Anonymous on Continually aim just beyond yo… ES on Heath-Brown’s theorem on… ES on Heath-Brown’s theorem on… Anonymous on The Collatz conjecture, Little… Hollis Williams on Homogenization of iterated sin…