Linear Diophantine equations in Piatetski-Shapiro sequences

Toshiki Matsusaka ¹   ; Kota Saito ²

¹ Institute for Advanced Research Nagoya University Furo-cho, Chikusa-ku Nagoya, 464-8602, Japan
² Graduate School of Mathematics Nagoya University Furo-cho, Chikusa-ku Nagoya, 464-8602, Japan

Acta Arithmetica, Tome 200 (2021) no. 1, pp. 91-110

Résumé

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 $3$.

Détail

DOI : 10.4064/aa200927-15-2

Keywords: piatetski shapiro sequence exponent alpha sequence integer parts alpha ldots non integral alpha mathrm alpha denote set those terms article study set alpha equation has infinitely many solutions mathrm alpha pairwise distinct lower bound its hausdorff dimension corollary uncountably many alpha mathrm alpha contains infinitely many arithmetic progressions length nbsp