Inhomogeneous Khintchine-Groshev theorem without monotonicity

TL;DR

This paper proves the inhomogeneous Khintchine-Groshev theorem without the monotonicity condition for specific cases, advancing understanding in Diophantine approximation theory.

Contribution

It confirms the conjecture for the case (2,1) and extends results to (1,2) with a rational inhomogeneous parameter, removing the monotonicity restriction.

Findings

Proved the conjecture for (n,m)=(2,1).

Extended the theorem to (n,m)=(1,2) with rational inhomogeneity.

Removed the monotonicity condition in these cases.

Abstract

The Khintchine-Groshev theorem in Diophantine approximation theory says that there is a dichotomy of the Lebesgue measure of sets of $\psi$-approximable numbers, given a monotonic function $\psi$. Allen and Ram\'irez removed the monotonicity condition from the inhomogeneous Khintchine-Groshev theorem for cases with $nm\geq3$ and conjectured that it also holds for $nm=2$. In this paper, we prove this conjecture in the case of $(n,m)=(2,1)$. We also prove it for the case of $(n,m)=(1,2)$ with a rational inhomogeneous parameter.