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 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.
