A Khintchine-type theorem and solutions to linear equations in Piatetski-Shapiro sequences

TL;DR

This paper extends Khintchine's theorem to a perturbed setting, establishing conditions under which Diophantine inequalities with shifting targets have infinitely many solutions, and applies this to solutions in Piatetski-Shapiro sequences.

Contribution

It introduces a novel perturbation framework allowing the perturbation magnitude to exceed the approximation threshold, and applies it to solutions of linear equations in Piatetski-Shapiro sequences.

Findings

Infinitely many solutions exist under new perturbation conditions.

Solutions in Piatetski-Shapiro sequences depend on the exponent f4f2.

The result characterizes solution counts based on f4f2.

Abstract

Our main result concerns a perturbation of a classic theorem of Khintchine in Diophantine approximation. We give sufficient conditions on a sequence of positive real numbers $(\psi_n)_{n \in \mathbb{N}}$ and differentiable functions $(\varphi_n: J \to \mathbb{R})_{n \in \mathbb{N}}$ so 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 of the perturbation $|\varphi_n(\theta)|$ 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. $\alpha > 1$, it has infinitely many or finitely many solutions of the form $\lfloor n^\alpha \rfloor$ according as $\alpha < 2$ or $\alpha > 2$.