Zero-one laws for uniform inhomogeneous Diophantine approximations

Abstract: In [Compositio Math. 155 (2019)] Kleinbock and Wadleigh proved a “zero-one law” for uniform Diophantine approximations to pairs (\Theta, \eta) of a matrix \Theta and vector \eta by using dynamics on the space of grids. We will show how the classical Diophantine transference principle provides an alternative approach to this problem and allows us to prove some generalizations. Namely, we will reduce the statement for pairs to the twisted (“fixed matrix”) case and show zero-one laws for twisted uniform approximations. All the proofs are made in weighted case and, more generally, in the setup of approximations with arbitrary weight functions, which will also be discussed. This talk is based on arXiv:2508.01912 and arXiv:2503.21180.

dynamical systemsnumber theory

Audience: general audience

New England Dynamics and Number Theory Seminar