Finiteness of solutions to linear Diophantine equations on Piatetski-Shapiro sequences
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866912579934224384 |
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| author | Saito, Kota |
| author_facet | Saito, Kota |
| 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 $\mathrm{PS}(α)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2306_17813 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Finiteness of solutions to linear Diophantine equations on Piatetski-Shapiro sequences Saito, Kota Number Theory 11D04, 11K55 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 $\mathrm{PS}(α)$. |
| title | Finiteness of solutions to linear Diophantine equations on Piatetski-Shapiro sequences |
| topic | Number Theory 11D04, 11K55 |
| url | https://arxiv.org/abs/2306.17813 |