Linear Diophantine equations in Piatetski-Shapiro sequences
Volume 200 / 2021
Acta Arithmetica 200 (2021), 91-110
MSC: Primary 11K55; Secondary 11D04.
DOI: 10.4064/aa200927-15-2
Published online: 10 May 2021
Abstract
The Piatetski-Shapiro sequence with exponent $\alpha $ is the sequence of integer parts of $n^\alpha $ $(n = 1,2,\ldots )$ with a non-integral $\alpha \gt 0$. We let $\mathrm {PS}(\alpha )$ denote the set of those terms. In this article, we study the set of $\alpha $ such that the equation $ax + by = cz$ has infinitely many solutions $(x,y,z) \in \mathrm {PS}(\alpha )^3$ with $x,y,z$ pairwise distinct, and give a lower bound for its Hausdorff dimension. As a corollary, we find uncountably many $\alpha \gt 2$ such that $\mathrm {PS}(\alpha )$ contains infinitely many arithmetic progressions of length $3$.
