23 research outputs found
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
On the maximal number of three-term arithmetic progressions in subsets of Z/pZ
Let a be a real number between 0 and 1. Ernie Croot showed that the quantity
\max_A #(3-term arithmetic progressions in A)/p^2, where A ranges over all
subsets of Z/pZ of size at most a*p, tends to a limit as p tends to infinity
through primes. Writing c(a) for this limit, we show that c(a) = a^2/2 provided
that a is smaller than some absolute constant. In fact we prove rather more,
establishing a structure theorem for sets having the maximal number of 3-term
progressions amongst all subsets of Z/pZ of cardinality m, provided that m <
c*p.Comment: 12 page
An improvement to the Kelley-Meka bounds on three-term arithmetic progressions
In a recent breakthrough Kelley and Meka proved a quasipolynomial upper bound
for the density of sets of integers without non-trivial three-term arithmetic
progressions. We present a simple modification to their method that strengthens
their conclusion, in particular proving that if has no
non-trivial three-term arithmetic progressions then for some .Comment: 9 page