Let SO(3,R) be the 3D-rotation group equipped with the real-manifold topology and the normalized Haar measure \mu. Confirming a conjecture by Breuillard and Green, we show that if A is an open subset of SO(3,R) with sufficiently small measure, then...

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Special Year Research Seminar

### Around Gauss Circle Problem: Hardy's Conjecture and the Distribution of Lattice Points Near Circles

This talk is based on a joint work with Steve Lester.

We review the Gauss circle problem, and Hardy's conjecture regarding the order of magnitude of the remainder term. It is attempted to rigorously formulate the folklore heuristics behind Hardy's...

Ruzsa asked whether there exist Fourier-uniform subsets of ℤ/Nℤ with very few 4-term arithmetic progressions (4-AP). The standard pedagogical example of a Fourier uniform set with a "wrong" density of 4-APs actually has 4-AP density much higher than...

I will discuss pointwise ergodic theory as it developed out of Bourgain's work in the 80s, leading up to my work with Mirek and Tao on bilinear ergodic averages.

In 1996 Manjul Barghava introduced a notion of P-orderings for arbitrary sets S of a Dedekind domain, with respect to a prime ideal P, which defined associated invariants called P-sequences. He combined these invariants to define generalized...

In its dynamical formulation, the Furstenberg—Sárközy theorem states that for any invertible measure-preserving system (X,μ,T), any set A⊆X with μ(A) greater than 0, and any integer polynomial P with P(0)=0,

c(A)=limN−M→∞1N−M∑n=MN−1μ(A∩TP(n)A)>0.

...Given any bounded multiplicative function a deep conjecture of Elliott predicts

cancellations of the form for all distinct shifts unless it is ``close" to the modulated Dirichlet character in an

appropriate sense. Partial progress towards this...

The goal of this talk is to present new results dealing with the asymptotic joint independence properties of commuting strongly mixing transformations along polynomials. These results form natural strongly mixing counterparts to various weakly and...

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...