Metric theory of inhomogeneous Diophantine approximations with a fixed matrix

TL;DR

The paper develops a metric theory for inhomogeneous Diophantine approximation with a fixed matrix $\Theta$, revealing deep dualities between uniform inhomogeneous approximations of $(\Theta,\pmb{\eta})$ and homogeneous approximations of the transposed matrix $\Theta^{\top}$. It proves analogues of Dirichlet and Davenport–Schmidt-type results in this fixed-matrix setting and shows that the set of singular vectors is winning while badly approximable shifts have measure-zero but often full Hausdorff dimension. The authors establish asymptotic and uniform variants of transference principles, derive Kurzweil-type convergence-divergence criteria, and revisit Khintchine-type theorems, including explicit special cases and corollaries. The work advances the metrical theory by linking inhomogeneous and homogeneous perspectives through transference, yielding zero-one laws, full-dimension results, and robust topological properties relevant to dynamics on lattices and Diophantine geometry.

Abstract

In this paper we develop a metric theory of inhomogeneous Diophantine approximation for the case of a fixed matrix. We use transference principle to connect uniform Diophantine properties of a pair $(Θ, \pmbη)$ of a matrix and a vector with the asymptotic Diophantine properties of the transposed matrix $Θ^{\top}$, and vice versa, the asymptotic Diophantine properties of a pair $(Θ, \pmbη)$ with asymptotic Diophantine properties of the transposed matrix. In these setups, we prove analogues of classical statements of metrical homogeneous Diophantine approximations and answer some open questions that were raised in recent works.

Theorems & Definitions (60)

• Definition 1.1
• Definition 1.2
• Theorem 1.3
• Theorem 1.4
• Definition 2.1
• Definition 2.2
• Definition 2.3
• Theorem A
• Theorem 3.1
• Theorem 3.2