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arXiv · 2508.01912

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

Vasiliy Neckrasov
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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.

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Vasiliy Neckrasov. 2025-08-03. Khintchine-type theorems for weighted uniform inhomogeneous approximations via transference principle. https://arxiv.org/abs/2508.01912

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