A version of the polynomial Szemer´edi theorem was shown to hold
in finite fields in [BLM05]. In particular, one has patterns ${x, x
+ P1(n), . . . , x + Pk(n)}$ (1) for polynomials with zero constant
term in large subsets of finite fields. When the...
The works of Furstenberg and Bergelson-Leibman on the Szemeredi
theorem and its polynomial extension motivated the study of the
limiting behavior of multiple ergodic averages of commuting
transformations with polynomial iterates. Following
important...
I will present results establishing cancellation in short sums
of arithmetic functions (in particular the von Mangoldt and divisor
functions) twisted by polynomial exponential phases, or more
general nilsequence phases. These results imply the...
We will discuss multilinear variants of Weyl's inequality for
the exponential sums arising in pointwise convergence problems
related to the Furstenberg-Bergelson-Leibman conjecture. We will
also illustrate how to use the multilinear Weyl inequality...
In this talk we present a natural generalization of a sumset
conjecture of Erdos to higher orders, asserting that every subset
of the integers with positive density contains a sumset $B_1+\ldots
+B_k$ where $B_1, \ldots , B_k$ are infinite. Our...
In the 1970’s Erdos asked several questions about what kind of
infinite structures can be found in every set of natural numbers
with positive density. In recent joint work with Kra, Richter and
Robertson we proved that every such set A can be...
For a topological dynamical systems (T, X) and a fixed $x \in X$
we are interested in the distribution of prime and semi-prime
orbits, i.e. ${T px}p='$ and ${T p1p2 x}p1,p2='$. We are interested
in systems for which the related sequences of...
Given $B \subset N$, we consider the corresponding set $FB$ of
$B$-free integers, i.e. $n \in FB i_ no b \in B$ divides $n$. We
$de_{ne} X \eta_}$ the B-free subshift _ as the smallest subshift
containing $\eta := 1FB \in {0, 1}Z$. Such systems are...
