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| Format: | Preprint |
| Published: |
2025
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| Online Access: | https://arxiv.org/abs/2504.08502 |
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| _version_ | 1866908480598704128 |
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| author | Santana, Sebastián Carrillo |
| author_facet | Santana, Sebastián Carrillo |
| contents | We develop a general approach for showing when a set of integers $\mathscr{A}$ has infinitely many $k^{th}$ powerfree numbers without relying on equidistribution estimates for $\mathscr{A}$. In particular, we show that if the Fourier transform of $\mathscr{A}$ satisfies certain $L^{\infty}$ and $L^{1}$ bounds, and is also "decreasing" in some sense, then $\mathscr{A}$ contains infinitely many $k^{th}$ powerfree numbers. We then use this method to show that there are infinitely many cubefree palindromes in base $b\ge 1100$, and in the process we obtain new $L^{1}$ bounds for the Fourier transform of the set of palindromes. We also show that there are infinitely many squarefree integers such that its reverse is also squarefree in any base $b\ge 2$. Moreover, we show that there are infinitely many squarefree integers with a missing digit in base $b\ge 5$, and infinitely many such cubefree integers in base $b\ge 3$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_08502 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Powerfree integers and Fourier bounds Santana, Sebastián Carrillo Number Theory We develop a general approach for showing when a set of integers $\mathscr{A}$ has infinitely many $k^{th}$ powerfree numbers without relying on equidistribution estimates for $\mathscr{A}$. In particular, we show that if the Fourier transform of $\mathscr{A}$ satisfies certain $L^{\infty}$ and $L^{1}$ bounds, and is also "decreasing" in some sense, then $\mathscr{A}$ contains infinitely many $k^{th}$ powerfree numbers. We then use this method to show that there are infinitely many cubefree palindromes in base $b\ge 1100$, and in the process we obtain new $L^{1}$ bounds for the Fourier transform of the set of palindromes. We also show that there are infinitely many squarefree integers such that its reverse is also squarefree in any base $b\ge 2$. Moreover, we show that there are infinitely many squarefree integers with a missing digit in base $b\ge 5$, and infinitely many such cubefree integers in base $b\ge 3$. |
| title | Powerfree integers and Fourier bounds |
| topic | Number Theory |
| url | https://arxiv.org/abs/2504.08502 |