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| Format: | Preprint |
| Published: |
2023
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2306.17813 |
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Table of Contents:
- A sequence of integers of the form $\lfloor n^α\rfloor$ $(n=1,2,\ldots)$ for some fixed non-integral $α>1$ is called a Piatetski-Shapiro sequence, where $\lfloor x\rfloor$ denotes the integer part of $x$. Let $\mathrm{PS}(α)$ denote the set of all those terms. In this article, we show that $x+y=z$ has only finitely many solutions $(x,y,z)\in \mathrm{PS}(α)^3$ for almost every $α>3$. Furthermore, we show that $\mathrm{PS}(α)$ has only finitely many arithmetic progressions of length $3$ for almost every $α>10$. In addition, we estimate upper bounds for the Hausdorff dimension of the set of $α\in [s,t]$ such that $y=a_1x_1+\cdots +a_nx_n$ has infinitely many solutions on $\mathrm{PS}(α)$.