Khintchine's theorem for inhomogeneous simultaneous approximation with polynomial decay

TL;DR

This paper extends Khintchine's theorem to inhomogeneous simultaneous approximation in higher dimensions, specifically resolving the (1,2) case under polynomial decay conditions.

Contribution

It proves the conjecture that the monotonicity condition can be removed in the (1,2) inhomogeneous case with polynomial decay.

Findings

Resolved the (1,2) case for polynomial decay functions.

Extended Khintchine's theorem to inhomogeneous higher-dimensional settings.

Confirmed conjecture about removing monotonicity in specific cases.

Abstract

Khintchine's theorem on the measure dichotomy for the set of $\psi$-approximable numbers has been generalized to inhomogeneous and higher-dimensional settings. Allen and Ram\'irez conjectured that the monotonicity condition can be removed in the inhomogeneous $nm=2$ cases. In this paper, we resolve the $(n,m)=(1,2)$ case for $\psi$ satisfying a polynomial decay condition $\psi(q)=O(q^{-\delta})$ for some $\delta>0.$