458 research outputs found
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
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
Roth's theorem for four variables and additive structures in sums of sparse sets
We show that if a subset A of {1,...,N} does not contain any solutions to the
equation x+y+z=3w with the variables not all equal, then A has size at most
exp(-c(log N)^{1/7}) N, where c > 0 is some absolute constant. In view of
Behrend's construction, this bound is of the right shape: the exponent 1/7
cannot be replaced by any constant larger than 1/2.
We also establish a related result, which says that sumsets A+A+A contain
long arithmetic progressions if A is a subset of {1,...,N}, or high-dimensional
subspaces if A is a subset of a vector space over a finite field, even if A has
density of the shape above.Comment: 23 page
Arithmetic structures in random sets
We extend two well-known results in additive number theory, S\'ark\"ozy's
theorem on square differences in dense sets and a theorem of Green on long
arithmetic progressions in sumsets, to subsets of random sets of asymptotic
density 0. Our proofs rely on a restriction-type Fourier analytic argument of
Green and Green-Tao.Comment: 22 page
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