We show that for every positive integer k there are positive
constants C and c such that if A is a subset of {1, 2, ..., n} of
size at least C n^{1/k}, then, for some d \leq k-1, the set of
subset sums of A contains a homogeneous d-dimensional...
The last decade has witnessed a revolution in the circle of
problems concerned with proving sharp moment inequalities for
exponential sums on tori. This has in turn led to a better
understanding of pointwise estimates, but this topic remains...
Incidence bound for points and spheres in higher dimensions
generally becomes trivial in higher dimensions due to the existence
of the Lenz example consisting of two orthogonal circles
in ℝ4, and the corresponding construction in higher
dimensions...
It is an open question as to whether the prime numbers contain
the sum A+B of two infinite sets of natural numbers A, B (although
results of this type are known assuming the Hardy-Littlewood prime
tuples conjecture). Using the Maynard sieve and the...
Finding the smallest integer N=ES_d(n) such that in every
configuration of N points in R^d in general position there exist n
points in convex position is one of the most classical problems in
extremal combinatorics, known as the Erdős-Szekeres...
The Chowla conjecture from 1965 predicts that all
autocorrelations of the Liouville function vanish. In fact, after
an adaptation, the Chowla conjecture was expected to hold for all
aperiodic multiplicative functions with values in the unit disc
(cf...
We discuss some recent progress on the model-theoretic problem
of classifying the reducts of the complex field (with named
parameters and up to interdefinability). The tools we use include
Castle’s recent solution of the Restricted Trichotomy...
A conjecture of Erdős states that for every large enough prime
q, every reduced residue class modulo q is the product of two
primes less than q. I will discuss my on-going work with Kaisa
Matomäki establishing among other things a ternary variant
of...
If a set of integers is syndetic (finitely many translates cover
the integers), must it contain two integers whose ratio is a
square? No one knows. In the broader context of the
disjointness between additive and multiplicative configurations
and...
The celebrated product theorem says if A is a generating subset
of a finite simple group of Lie type G, then |AAA| \gg \min \{
|A|^{1+c}, |G| \}. In this talk, I will show that a similar
phenomenon appears in the continuous setting: If A is a
subset...
