1,929 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
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
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
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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