A Khintchinetype theorem and solutions to linear equations in PiatetskiShapiro sequences
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 Lebesguea.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 Lebesguea.e. $\alpha > 1$, it has infinitely many or finitely many solutions of the form $\lfloor n^\alpha \rfloor$ according as $\alpha < 2$ or $\alpha > 2$.
 Publication:

arXiv eprints
 Pub Date:
 September 2018
 arXiv:
 arXiv:1809.00360
 Bibcode:
 2018arXiv180900360G
 Keywords:

 Mathematics  Number Theory
 EPrint:
 18 pages