A height gap theorem for coefficients of Mahler functions
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arXiv
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| Format: | Preprint |
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2020
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| _version_ | 1866917195165990912 |
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| author | Adamczewski, Boris Bell, Jason Smertnig, Daniel |
| author_facet | Adamczewski, Boris Bell, Jason Smertnig, Daniel |
| contents | We study the asymptotic growth of coefficients of Mahler power series with algebraic coefficients, as measured by their logarithmic Weil height. We show that there are five different growth behaviors, all of which being reached. Thus, there are \emph{gaps} in the possible growths. In proving this height gap theorem, we obtain that a $k$-Mahler function is $k$-regular if and only if its coefficients have height in $O(\log n)$. Furthermore, we deduce that, over an arbitrary ground field of characteristic zero, a $k$-Mahler function is $k$-automatic if and only if its coefficients belong to a finite set. As a by-product of our results, we also recover a conjecture of Becker which was recently settled by Bell, Chyzak, Coons, and Dumas. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2003_03429 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | A height gap theorem for coefficients of Mahler functions Adamczewski, Boris Bell, Jason Smertnig, Daniel Number Theory We study the asymptotic growth of coefficients of Mahler power series with algebraic coefficients, as measured by their logarithmic Weil height. We show that there are five different growth behaviors, all of which being reached. Thus, there are \emph{gaps} in the possible growths. In proving this height gap theorem, we obtain that a $k$-Mahler function is $k$-regular if and only if its coefficients have height in $O(\log n)$. Furthermore, we deduce that, over an arbitrary ground field of characteristic zero, a $k$-Mahler function is $k$-automatic if and only if its coefficients belong to a finite set. As a by-product of our results, we also recover a conjecture of Becker which was recently settled by Bell, Chyzak, Coons, and Dumas. |
| title | A height gap theorem for coefficients of Mahler functions |
| topic | Number Theory |
| url | https://arxiv.org/abs/2003.03429 |