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arXiv · 2009.12899

Linear Diophantine equations in Piatetski-Shapiro sequences

Toshiki Matsusaka·Kota Saito
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Abstract

A Piatetski-Shapiro sequence with exponent $\alpha$ is a sequence of integer parts of $n^\alpha$ $(n = 1,2,\ldots)$ 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 $3$.

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Toshiki Matsusaka, Kota Saito. 2020-09-27. Linear Diophantine equations in Piatetski-Shapiro sequences. https://doi.org/10.4064/aa200927-15-2

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