A perturbed Khinchin-type theorem and solutions to linear equations in Piatetski-Shapiro sequences
Volume 192 / 2020
Abstract
Our main result concerns a perturbation of a classic theorem of Khinchin in Diophantine approximation. We give sufficient conditions on a sequence of positive real numbers $(\psi _n)_{n \in \mathbb N }$ and differentiable functions $(\varphi _n: J \to \mathbb R )_{n \in \mathbb N }$ such that for Lebesgue-a.e. $\theta \in J$, the inequality $\| n\theta + \varphi _n(\theta ) \| \leq \psi _n$ has infinitely many solutions. The main novelty is that the magnitude $|\varphi _n(\theta )|$ of the perturbation is allowed to exceed $\psi _n$, changing the usual “shrinking targets” problem into a “shifting targets” problem. As an application of the main result, we prove that if the linear equation $y=ax+b$, $a, b \in \mathbb R $, has infinitely many solutions in $\mathbb N $, then for Lebesgue-a.e. $\alpha \gt 1$, it has infinitely many or finitely many solutions of the form $\lfloor n^\alpha \rfloor $ according as $\alpha \lt 2$ or $\alpha \gt 2$.
