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

Abstract

Sz{ü}sz's inhomogeneous version (1958) of Khintchine's theorem (1924) gives conditions on $ψ:\mathbb{N}\to\mathbb{R}_{\geq 0}$ under which for almost every real number $α$ there exist infinitely many rationals $p/q$ such that \begin{equation*} \lvertα- \frac{p+γ}{q}\rvert < \frac{ψ(q)}{q}, \end{equation*} where $γ\in\mathbb{R}$ is some fixed inhomogeneous parameter. It is often interpreted as a statement about visits of $qα\,(\bmod 1)$ to a shrinking target centered around $γ\,(\bmod 1)$, viewed in $\mathbb{R}/\mathbb{Z}$. Hauke and the second author have conjectured that Sz{ü}sz's result continues to hold if the target is allowed to move as well as shrink, that is, if the inhomogeneous parameter $γ$ is allowed to depend on the denominator $q$ of the approximating rational. We show that the conjecture holds under an ``extra divergence'' assumption on $ψ$. 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.