Solving xz=yy in certain subsets of finite groups
We show that if G is a finite group and A is a subset of G with no non-trivial solutions to xz=yy then |A| < |G|/(log log |G|)^c.
math.CA↗
arXiv subjects
Tom Sanders
Publications and source records attributed to Tom Sanders.
A statistical approach to covering lemmas
We discuss a statistical variant of Ruzsa's covering lemma and use it to show that if G is an Abelian group of bounded exponent and A in G has |A+A| < K|A| then the subgroup generated by A has size at most exp(O(K log^22K))|A|, where the constant in the big-O depends on the exponent of the group only.
math.CO↗
Fourier uniformity on subspaces
Let $\mathbb{F}$ be a fixed finite field, and let $A \subset \mathbb{F}^n$. It is a well-known fact that there is a subspace $V \leq \mathbb{F}^n$, $\mbox{codim} V \ll_δ 1$, and an $x$, such that $A$ is $δ$-uniform when restricted to $x + V$ (that is, all non-trivial Fourier coefficients of $A$ restricted to $x + V$ have magnitude at most $δ$). We show that if $\mathbb{F} = \mathbb{F}_2$ then it is possible to take $x = 0$; that is, $A$ is $δ$-uniform on a subspace $V \leq \mathbb{F}^n$. We give an example to show that this is not necessarily possible when $\mathbb{F} = \mathbb{F}_3$. ADDED July 2016: shortly after this paper appeared on the arxiv, F. Manners showed us a rather short argument he had found in 2013, giving a better bound for our main theorem. We do not, therefore, intend to publish this note. The example over $\mathbb{F}_3$ may still be of interest to some readers and so we will not withdraw the paper from the arxiv.
math.NT↗
Approximate (Abelian) groups
ECM survey article discussing the structure of subsets of Abelian groups which behave `a bit like' cosets (of subgroups).
math.CA↗
An analytic approach to a weak non-Abelian Kneser-type theorem
We prove the following result due to Hamidoune using an analytic approach. Suppose that A is a subset of a finite group G with |AA^{-1}| \leq (2-\varepsilon)|A|. Then there is a subgroup H of G and a set X of size O_\varepsilon(1) such that A \subset XH.
math.CA↗
Approximate groups and doubling metrics
We develop a version of Freiman's theorem for a class of non-abelian groups, which includes finite nilpotent, supersolvable and solvable A-groups. To do this we have to replace the small doubling hypothesis with a stronger relative polynomial growth hypothesis akin to that in Gromov's theorem (although with an effective range), and the structures we find are balls in (left and right) translation invariant pseudo-metrics with certain well behaved growth estimates. Our work complements three other recent approaches to developing non-abelian versions of Freiman's theorem by Breuillard and Green, Fischer, Katz and Peng, and Tao.
math.CA↗
The structure theory of set addition revisited
In this article we survey some of the recent developments in the structure theory of set addition.
math.CA↗
On the Bogolyubov-Ruzsa lemma
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|.
math.CA↗
On Roth's theorem on progressions
We show that if A is a subset of {1,...,N} contains no non-trivial three-term arithmetic progressions then |A|=O(N/ log^{1-o(1)} N). The approach is somewhat different from that used in arXiv:1007.5444.
math.CA↗
Structure in sets with logarithmic doubling
Suppose that G is an abelian group, A is a finite subset of G with |A+A|< K|A| and eta in (0,1] is a parameter. Our main result is that there is a set L such that |A cap Span(L)| > K^{-O_eta(1)}|A| and |L| = O(K^eta log |A|). We include an application of this result to a generalisation of the Roth-Meshulam theorem due to Liu and Spencer.
math.CA↗
On certain other sets of integers
We show that if A is a subset of {1,...,N} containing no non-trivial three-term arithmetic progressions then |A|=O(N/ log^{3/4-o(1)} N).
math.NT↗
On a theorem of Shkredov
We show that if A is a finite subset of an abelian group with additive energy at least c|A|^3 then there is a subset L of A with |L|=O(c^{-1}\log |A|) such that |A \cap Span(L)| >> c^{1/3}|A|.
math.CA↗
Popular difference sets
We provide further explanation of the significance of a construction in a recent paper of Wolf [Israel J. Math. 179 (2010), 253-278] in the context of the problem of finding large subspaces in sumsets.
math.CO↗
Green's sumset problem at density one half
We investigate the size of subspaces in sumsets and show two main results. First, if A is a subset of F_2^n with density at least 1/2 - o(n^{-1/2}) then A+A contains a subspace of co-dimension 1. Secondly, if A is a subset of F_2^n with density at least 1/2-o(1) then A+A contains a subspace of co-dimension o(n).
math.NT↗
Appendix to 'Roth's theorem on progressions revisited' by J Bourgain
We show two results. First, a refinement of Freiman's theorem: if A is a finite set of integers and |A+A| < K|A|, then A is contained in a multidimensional progression of dimension at most O(K^{7/4} log^3K) and size at most exp(O(K^{7/4} log^3K))|A|. Secondly, an improvement of a result of Konyagin and Laba: if A is a finite set of reals and a is a transcendental then |A+aA| >> |A|(log |A|)^{4/3-ε} for all ε>0.
math.CA↗
Chowla's cosine problem
Suppose that G is a discrete abelian group and A is a finite symmetric subset of G. We show two main results. i) Either there is a set H of O(log^c|A|) subgroups of G with |A \triangle \bigcup H| = o(|A|), or there is a character X on G such that -wh{1_A}(X) >> log^c|A|. ii) If G is finite and |A|>> |G| then either there is a subgroup H of G such that |A \triangle H| = o(|A|), or there is a character X on G such that -wh{1_A}(X)>> |A|^c.
math.CA↗
On a non-abelian Balog-Szemeredi-type lemma
We show that if G is a group and A is a finite subset of G with |A^2| < K|A|, then for all k there is a symmetric neighbourhood of the identity S with S^k a subset of A^2A^{-2} and |S| > exp(-K^{O(k)})|A|.
math.CO↗
A Freiman-type theorem for locally compact abelian groups
We prove a Freiman-type theorem for locally compact abelian groups. If A is a subset of a locally compact abelian group with Haar measure m and m(nA) < n^dm(A) for all n>d log d then we describe A in a way which is tight up to logarithmic factors in d.
math.CA↗