Dynamical Systems Seminar

Given \(x\) in \([0,1]^d\), this talk is about the fine-scale distribution of the Kronecker sequence \((n x \text{ mod } 1)_{n\geq 1}\). After a general introduction, I will report on forthcoming work with Sam Chow. We establish a novel deterministic analogue of Beck’s local-to-global principle (Ann. of Math. 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.