Erdős-style geometry is concerned with combinatorial questions
about simple geometric objects, such as counting incidences between
finite sets of points, lines, etc. These questions can be typically
viewed as asking for the possible number of...
Given n∈ℕ and ξ∈ℝ, let τ(n;ξ)=∑d|ndiξ. Hall and Tenenbaum asked
in their book \textit{Divisors} what is the value of
maxξ∈[1,2]|τ(n;ξ)| for a ``typical'' integer n. I will present work
in progress, carried out in collaboration with Louis-Pierre...
Complex dynamics explores the evolution of points under
iteration of functions of complex variables. In this talk I will
introduce into the context of complex dynamics, a new approximation
tool allowing us to construct new examples of entire...
The Mackey-Zimmer representation theorem is a key structural
result from ergodic theory: Every compact extension between ergodic
measure-preserving systems can be written as a skew-product by a
homogeneous space of a compact group. This is used, e.g...
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...
