Fall 2024 Titles and Abstracts – New England Dynamics and Number Theory Seminar

Date: 24 September 2024
Speaker: Sam Chow
Title: Smooth discrepancy and Littlewood’s conjecture
Abstract: Given $\boldsymbol \alpha \in [0,1]^d$ , we estimate the smooth discrepancy of the Kronecker sequence $(n \boldsymbol \alpha \: \mathrm{mod} \: 1)_{n=1}^\infty$ . We find that it can be smaller than the classical discrepancy of any sequence when $d \le 2$ , and can even be bounded in the case $d=1$ . To achieve this, we establish a novel deterministic analogue of Beck’s local-to-global principle (Annals 1994), which relates the discrepancy of a Kronecker sequence to multiplicative diophantine approximation. This opens up a new avenue of attack for Littlewood’s conjecture.

Date: 8 October 2024
Speaker: Manuel Hauke
Title: Metric Diophantine approximation: Moving targets and inhomogeneous variants
Abstract: Khintchine’s Theorem and its inhomogeneous and multidimensional variants provide a satisfying answer about the quality of approximations for almost every number. In this talk, I will discuss the (still open) question of allowing a moving target (that is, the inhomogeneous parameter changes for each denominator) in Khintchine’s Theorem. Furthermore, I will describe Duffin–Schaeffer-type results and conjectures in these setups, both in dimension 1, but also in higher dimensions. This is partially joint work with Victor Beresnevich and Sanju Velani, respectively with Felipe Ramírez.