Thin subbases of Piatetski-Shapiro sequences
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866918485853995008 |
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| author | Táfula, Christian |
| author_facet | Táfula, Christian |
| contents | For a non-integral real number $c>1$, let $\mathbb{N}_{(c)}:=\{\lfloor n^c\rfloor ~|~ n\in\mathbb{N}\}$. We show that $\mathbb{N}_{(c)}$ contains thin subbases of every order $h\geq 5$ when $1<c<2$, and $h\geq (\lfloor 2c\rfloor+1)(\lfloor 2c\rfloor+2)+1$ when $c>2$. In fact, for every regularly varying function $F$ such that \[ \frac{F(x)}{\log x}\to\infty\quad\text{ and } \quad F(x)\leq (1+o(1))\frac{Γ(1+1/c)^h}{Γ(h/c)} x^{h/c-1}, \] there exists $A\subseteq\mathbb{N}_{(c)}$ with $r_{A,h}(n)\sim F(n)$. We also establish analogous results for $k$-th powers of Piatetski-Shapiro numbers and Piatetski-Shapiro primes for small $c$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_04411 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Thin subbases of Piatetski-Shapiro sequences Táfula, Christian Number Theory Primary 11B13, 11B34, Secondary 11P05, 11P32, 11B83 For a non-integral real number $c>1$, let $\mathbb{N}_{(c)}:=\{\lfloor n^c\rfloor ~|~ n\in\mathbb{N}\}$. We show that $\mathbb{N}_{(c)}$ contains thin subbases of every order $h\geq 5$ when $1<c<2$, and $h\geq (\lfloor 2c\rfloor+1)(\lfloor 2c\rfloor+2)+1$ when $c>2$. In fact, for every regularly varying function $F$ such that \[ \frac{F(x)}{\log x}\to\infty\quad\text{ and } \quad F(x)\leq (1+o(1))\frac{Γ(1+1/c)^h}{Γ(h/c)} x^{h/c-1}, \] there exists $A\subseteq\mathbb{N}_{(c)}$ with $r_{A,h}(n)\sim F(n)$. We also establish analogous results for $k$-th powers of Piatetski-Shapiro numbers and Piatetski-Shapiro primes for small $c$. |
| title | Thin subbases of Piatetski-Shapiro sequences |
| topic | Number Theory Primary 11B13, 11B34, Secondary 11P05, 11P32, 11B83 |
| url | https://arxiv.org/abs/2605.04411 |