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

TL;DR

This paper extends zero-one laws for weighted uniform inhomogeneous Diophantine approximations, providing new proofs and characterizations of approximation sets using transference principles and measure theory.

Contribution

It generalizes existing zero-one laws to include arbitrary weights and offers a new proof based on transference, along with a complete measure-theoretic description of approximation sets.

Findings

Established a generalized zero-one law for weighted inhomogeneous approximations.

Provided a measure-theoretic characterization of $g$-Dirichlet pairs with fixed matrices.

Characterized badly approximable matrices via inhomogeneous approximation properties.

Abstract

In [Compositio Math. 155 (2019)] Kleinbock and Wadleigh proved a "zero-one law" for uniform inhomogeneous Diophantine approximations. We generalize this statement with arbitrary weight functions and establish a new and simple proof of this statement, based on transference principle. We also give a complete description of the sets of $g$-Dirichlet pairs with a fixed matrix in this setup 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.