56,998 research outputs found
A quantitative improvement for Roth's theorem on arithmetic progressions
We improve the quantitative estimate for Roth's theorem on three-term
arithmetic progressions, showing that if contains no
non-trivial three-term arithmetic progressions then . By the same method we also improve the bounds in the
analogous problem over and for the problem of finding long
arithmetic progressions in a sumset
A probabilistic technique for finding almost-periods of convolutions
We introduce a new probabilistic technique for finding 'almost-periods' of
convolutions of subsets of groups. This gives results similar to the
Bogolyubov-type estimates established by Fourier analysis on abelian groups but
without the need for a nice Fourier transform to exist. We also present
applications, some of which are new even in the abelian setting. These include
a probabilistic proof of Roth's theorem on three-term arithmetic progressions
and a proof of a variant of the Bourgain-Green theorem on the existence of long
arithmetic progressions in sumsets A+B that works with sparser subsets of {1,
..., N} than previously possible. In the non-abelian setting we exhibit
analogues of the Bogolyubov-Freiman-Halberstam-Ruzsa-type results of additive
combinatorics, showing that product sets A B C and A^2 A^{-2} are rather
structured, in the sense that they contain very large iterated product sets.
This is particularly so when the sets in question satisfy small-doubling
conditions or high multiplicative energy conditions. We also present results on
structures in product sets A B. Our results are 'local' in nature, meaning that
it is not necessary for the sets under consideration to be dense in the ambient
group. In particular, our results apply to finite subsets of infinite groups
provided they 'interact nicely' with some other set.Comment: 29 pages, to appear in GAF
Some new inequalities in additive combinatorics
In the paper we find new inequalities involving the intersections of shifts of some subset from an abelian group. We apply the
inequalities to obtain new upper bounds for the additive energy of
multiplicative subgroups and convex sets and also a series another results on
the connection of the additive energy and so--called higher moments of
convolutions. Besides we prove new theorems on multiplicative subgroups
concerning lower bounds for its doubling constants, sharp lower bound for the
cardinality of sumset of a multiplicative subgroup and its subprogression and
another results.Comment: 39 page
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|.Comment: 5 pp. Corrected typos. Minor revision
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