329 research outputs found
On sumsets of convex sets
A set of reals A={a_1,...,a_2} is called convex if a_{i+1} - a_i > a_i -
a_{i-1} for all i. We prove, in particular, that |A-A| \gg |A|^{8/5} \log{-2/5}
|A|.Comment: 6 page
On sums of Szemer\'edi--Trotter sets
We prove new general results on sumsets of sets having Szemer\'edi--Trotter
type. This family includes convex sets, sets with small multiplicative
doubling, images of sets under convex/concave maps and others.Comment: 12 page
John-type theorems for generalized arithmetic progressions and iterated sumsets
A classical theorem of Fritz John allows one to describe a convex body, up to
constants, as an ellipsoid. In this article we establish similar descriptions
for generalized (i.e. multidimensional) arithmetic progressions in terms of
proper (i.e. collision-free) generalized arithmetic progressions, in both
torsion-free and torsion settings. We also obtain a similar characterization of
iterated sumsets in arbitrary abelian groups in terms of progressions, thus
strengthening and extending recent results of Szemer\'edi and Vu.Comment: 20 pages, no figures, to appear, Adv. in Math. Some minor changes
thanks to referee repor
Compressions, convex geometry and the Freiman-Bilu theorem
We note a link between combinatorial results of Bollob\'as and Leader
concerning sumsets in the grid, the Brunn-Minkowski theorem and a result of
Freiman and Bilu concerning the structure of sets of integers with small
doubling.
Our main result is the following. If eps > 0 and if A is a finite nonempty
subset of a torsion-free abelian group with |A + A| <= K|A|, then A may be
covered by exp(K^C) progressions of dimension [log_2 K + eps] and size at most
|A|.Comment: 9 pages, slight revisions in the light of comments from the referee.
To appear in Quarterly Journal of Mathematics, Oxfor
Convexity and a sum-product type estimate
In this paper we further study the relationship between convexity and
additive growth, building on the work of Schoen and Shkredov (\cite{SS}) to get
some improvements to earlier results of Elekes, Nathanson and Ruzsa
(\cite{ENR}). In particular, we show that for any finite set
and any strictly convex or concave function ,
and For the latter of
these inequalities, we go on to consider the consequences for a sum-product
type problem
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