Quantitative inhomogeneous Diophantine approximation for systems of linear forms

TL;DR

This paper proves that for systems of linear forms with dimensions (1,2), the monotonicity condition in inhomogeneous Diophantine approximation can be removed when the parameter is a non-Liouville irrational, and provides new asymptotic formulas.

Contribution

It confirms the conjecture that monotonicity is unnecessary for (1,2) systems in inhomogeneous approximation and introduces an asymptotic formula with near square-root cancellation.

Findings

Monotonicity can be removed for (1,2) systems with non-Liouville irrationals.

Established an asymptotic formula with almost square-root cancellation.

Refined overlap estimates applicable to inhomogeneous Duffin-Schaeffer conjecture.

Abstract

The inhomogeneous Khintchine-Groshev Theorem is a classical generalization of Khintchine's Theorem in Diophantine approximation, by approximating points in $\mathbb{R}^m$ by systems of linear forms in $n$ variables. Analogous to the question considered by Duffin and Schaeffer for Khintchine's Theorem (which is the case $m = n = 1$), the question arises for which $m,n$ the monotonicity can be safely removed. If $m = n = 1$, it is known that monotonicity is needed. Recently, Allen and Ramirez showed that for $mn \geq 3$, the monotonicity assumption is unnecessary, conjecturing this to also hold when $mn = 2$. In this article, we confirm this conjecture for the case $(m,n)=(1,2)$ whenever the inhomogeneous parameter is a non-Liouville irrational number. Furthermore, under mild assumptions on the approximation function, we show an asymptotic formula (with almost square-root cancellation), which is not even known for homogeneous approximation. The proof makes use of refined overlap estimates in the 1-dimensional setting, which may have other applications including the inhomogeneous Duffin-Schaeffer conjecture.