• Corpus ID: 119174223

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

@article{Glasscock2018AKT,
title={A Khintchine-type theorem and solutions to linear equations in Piatetski-Shapiro sequences},
author={Daniel Glasscock},
journal={arXiv: Number Theory},
year={2018}
}
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 $|\varphi_n(\theta… ## References SHOWING 1-5 OF 5 REFERENCES Solutions to certain linear equations in Piatetski-Shapiro sequences Denote by$\text{PS}(\alpha)$the image of the Piatetski-Shapiro sequence$n \mapsto \lfloor n^{\alpha} \rfloor$where$\alpha > 1$is non-integral and$\lfloor x \rfloor$is the integer part of$x
Metric number theory
Introduction 1. Normal numbers 2. Diophantine approximation 3. GCD sums with applications 4. Schmidt's method 5. Uniform distribution 6. Diophantine approximation with restricted numerator and