Khintchine-type theorems for weighted uniform inhomogeneous approximations via transference principle

Abstract

In 2019 Kleinbock and Wadleigh proved a “zero-one law” for uniform inhomogeneous Diophantine approximations. We generalize this statement to arbitrary weight functions and establish a new and simple proof of this statement, based on the transference principle. We also give a complete description of the sets of $$-Dirichlet pairs with a fixed matrix in this setthe up from Lebesgue measure point of view. As an application, we consider the set of badly approximable matrices and give a characterization of bad approximability in terms of inhomogeneous approximations. All the aforementioned metrical descriptions work (and sometimes can be strengthened) for weighted Diophantine approximations.