Tom Sanders

On Roth's theorem on progressions

We show that if Af 1;:::;Ng contains no nontrivial three-term arithmetic progressions thenjAj = O(N=log 1 o(1) N).

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|.

Behavioural management of antipsychotic‐induced weight gain: a review

Objective: Although psychiatrists are aware of weight gain induced by atypical antipsychotics, only few studies on behavioural interventions in this patient group are published. This review aims to summarize the evidence on effectiveness of behavioural interventions for weight gain in the general population and in‐patients treated with atypical antipsychotics.

The structure theory of set addition revisited

In this article we survey some of the recent developments in the structure theory of set addition.

Additive structures in sumsets

Suppose that A is a subset of the integers {1,...,N} of density a. We provide a new proof of a result of Green which shows that A+A contains an arithmetic progression of length exp(ca(log N)^{1/2}) for some absolute c>0. Furthermore we improve the length of progression guaranteed in higher sumsets; for example we show that A+A+A contains a progression of length roughly N^{ca} improving on the previous best of N^{ca^{2+\epsilon}}.

On certain other sets of integers

We show that if A ⊂ {1,...,N} contains no non-trivial three-term arithmetic progressions then |A| = O(N/log3/4−o(1)N).

A quantitative version of the idempotent theorem in harmonic analysis

Suppose that G is a locally compact abelian group, and write M(G) for the algebra of bounded, regular, complex-valued measures under convolution. A measure // G M(G) is said to be idempotent if /x*/i = //, or alternatively if j5 takes only the values 0 and 1. The Cohen-Helson-Rudin idempotent theorem states that a measure \x is idempotent if and only if the set {7 G G : i2(7) = 1} belongs to the coset ring of G, that is to say we may write L i=i

Boolean Functions with small Spectral Norm

Abstract.Let

The Littlewood-Gowers problem

f : {\mathbb{F}}^{n}_{2} \rightarrow \{0, 1\}