• 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… 

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