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

Volume 192 / 2020

Daniel Glasscock Acta Arithmetica 192 (2020), 267-288 MSC: Primary 11J83; Secondary 11B83. DOI: 10.4064/aa180905-22-7 Published online: 22 November 2019


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 $\alpha \gt 2$.


  • Daniel GlasscockMathematical Sciences Department
    University of Massachusetts Lowell
    Olney Science Center, 428J
    1 University Avenue
    Lowell, MA 01854, U.S.A.

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