An inhomogeneous Dirichlet theorem via shrinking targets

TL;DR

This paper establishes an integrability criterion for a Diophantine approximation problem involving inhomogeneous systems, linking it to a shrinking target problem on grid spaces, and discusses related open problems.

Contribution

It introduces a new integrability criterion for inhomogeneous Diophantine approximation via shrinking targets, extending understanding of solvability conditions for almost all pairs.

Findings

Provides an integrability criterion for inhomogeneous approximation systems.

Reduces the problem to a shrinking target problem on grid spaces.

Comments on the open homogeneous case for general dimensions.

Abstract

We give an integrability criterion on a real-valued non-increasing function $\psi$ guaranteeing that for almost all (or almost no) pairs $(A, \textbf{b})$, where $A$ is a real $m\times n$ matrix and $\textbf{b} \in \mathbb{R}^m$, the system $\|A \textbf{q}+\textbf{b}-\textbf{p}\|^m< \psi({T})$, $\|\textbf{q}\|^n<{T}$ is solvable in $\textbf{p} \in \mathbb{Z}^m$, $\textbf{q} \in \mathbb{Z}^n$ for all sufficiently large $T$. The proof consists of a reduction to a shrinking target problem on the space of grids in $\mathbb{R}^{m+n}$. We also comment on the homogeneous counterpart to this problem, whose $m=n=1$ case was recently solved, but whose general case remains open.