2,267 research outputs found
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|.Comment: 28 pp. Corrected typos. Added appendix on model settin
The Erdos-Moser sum-free set problem
We show that if A is a finite set of integers then it has a subset S of size
\log^{1+c} |A| (c>0 absolute) such that s+s' is never in A when s and s' are
distinct elements of S.Comment: 47 pages. Corrections and clarification
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.Comment: 21 pp. Corrected typos. Minor revision
Three-term arithmetic progressions and sumsets
Suppose that G is an abelian group and A is a finite subset of G containing
no three-term arithmetic progressions. We show that |A+A| >> |A|(log
|A|)^{1/3-\epsilon} for all \epsilon>0.Comment: 20 pp. Corrected typos. Updated references. Corrected proof of
Theorem 5.1. Minor revisions
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}}.Comment: 28 pp. Corrected typos. Updated references
The structure theory of set addition revisited
In this article we survey some of the recent developments in the structure
theory of set addition.Comment: 38p
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).Comment: 10 pp. Corrected typo
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