The inhomogeneous Khintchine Theorem in dimension two

TL;DR

This paper proves the inhomogeneous Khintchine Theorem in two dimensions without monotonicity, completing the metric theory of inhomogeneous Diophantine approximation and confirming a longstanding conjecture.

Contribution

It establishes the inhomogeneous Khintchine Theorem in dimension two without the monotonicity assumption, resolving a key open problem.

Findings

The inhomogeneous Khintchine Theorem holds in dimension 2 without monotonicity.

Confirms the last remaining case in the metric theory of inhomogeneous Diophantine approximation.

Settles a Khintchine--Groshev-type conjecture for systems of linear forms.

Abstract

We prove that the inhomogeneous variant of Khintchine's Theorem holds in dimension $2$ without any monotonicity assumption. This resolves the last remaining case in the metric theory of inhomogeneous Diophantine approximation: while the monotonicity assumption is known to be unnecessary in dimensions $m\geq 3$ and necessary in dimension $m=1$, the two-dimensional case has remained open. It also settles the final outstanding case of a Khintchine--Groshev-type theorem for the approximation of systems of linear forms, confirming a conjecture of the first and third authors. Our results bring the inhomogeneous theory of metric Diophantine approximation into alignment with its homogeneous counterpart.