Quantitative inhomogeneous Diophantine approximation for systems of linear forms

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.