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In unpublished lecture notes, William A. Veech considered the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...
Behaviors of objects of algebraic interest, such as polynomials,
elliptic curves and number fields -- many of which are still
unknown -- fall into the field of arithmetic statistics.. By
bringing in Fourier analysis to interplay with a wide
variety...
We discuss some consequences of the existence of a Siegel zero
for various questions relating to the distribution of the prime
numbers, and in particular to conjectures of Hardy-Littlewood and
Chowla type. This is joint work with Joni Teravainen.
Abstract: Low-lying horocycles are known to equidistribute on
the modular curve. Here we consider the joint distribution of two
low-lying horocycles of different speeds in the product of two
modular curves and show equidistribution under certain...
