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...
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...
In unpublished lecture notes, William A. Veech considered the
following potential property of the Möbius function:
"In any Furstenberg system of the Möbius function, the
zero-coordinate is orthogonal to any function measurable with
respect to the...
We construct examples showing that the correlation in the Mobius
disjointness conjecture can go to zero arbitrarily slowly. In fact,
our methods yield a more general result, where in lieu of μ(n) one
can put any bounded sequence such that the Cesàro...
I will discuss work in progress with Morgan Weiler on knot
filtered embedded contact homology (ECH) of open book
decompositions of S^3 along T(2,q) torus knots to deduce
information about the dynamics of symplectomorphisms of the genus
(q-1)/2 pages...
