Toward Khintchine's theorem with a moving target: extra divergence or finitely centered target

TL;DR

This paper extends Khintchine's theorem to cases where the inhomogeneous target varies with the denominator, proving the conjecture under certain divergence conditions and finite target sets, with implications for rational approximation coloring.

Contribution

It proves the conjecture that Khintchine's theorem holds for moving targets under extra divergence and finite target constraints, advancing inhomogeneous Diophantine approximation theory.

Findings

The conjecture holds under an extra divergence condition on .

The conjecture is valid when the inhomogeneous parameter varies over a finite set.

A finite-colorings version of the inhomogeneous Khintchine theorem is established.

Abstract

Sz{\"u}sz's inhomogeneous version (1958) of Khintchine's theorem (1924) gives conditions on $\psi:\mathbb{N}\to\mathbb{R}_{\geq 0}$ under which for almost every real number $\alpha$ there exist infinitely many rationals $p/q$ such that \begin{equation*} \lvert\alpha - \frac{p+\gamma}{q}\rvert < \frac{\psi(q)}{q}, \end{equation*} where $\gamma\in\mathbb{R}$ is some fixed inhomogeneous parameter. It is often interpreted as a statement about visits of $q\alpha\,(\bmod 1)$ to a shrinking target centered around $\gamma\,(\bmod 1)$, viewed in $\mathbb{R}/\mathbb{Z}$. Hauke and the second author have conjectured that Sz{\"u}sz's result continues to hold if the target is allowed to move as well as shrink, that is, if the inhomogeneous parameter $\gamma$ is allowed to depend on the denominator $q$ of the approximating rational. We show that the conjecture holds under an ``extra divergence'' assumption on $\psi$. We also show that it holds when the inhomogeneous parameter's movement is constrained to a finite set. As a byproduct, we obtain a finite-colorings version of the inhomogeneous Khintchine theorem, giving rational approximations with monochromatic denominators.