Linear Diophantine equations in Piatetski-Shapiro sequences

Abstract

A Piatetski-Shapiro sequence with exponent $\alpha$ is a sequence of integer parts of $n^{\alpha }$ $(n=1,2,\dots )$ with a non-integral $\alpha >0$. We let $\mathrm{PS}(\alpha )$ denote the set of those terms. In this article, we study the set of $\alpha$ so that the equation $ax+by=cz$ has infinitely many pairwise distinct solutions $(x,y,z)\in \mathrm{PS}(\alpha {)}^3$, and give a lower bound for its Hausdorff dimension. As a corollary, we find uncountably many $\alpha >2$ such that $\mathrm{PS}(\alpha )$ contains infinitely many arithmetic progressions of length $3$.

arXiv comment21 pages

References in corpus (1)

• Arithmetic Progressions in the Graphs of Slightly Curved Sequences

Cited by in corpus (1)

• On the number of representations of integers as differences between Piatetski-Shapiro numbers