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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2023
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2304.10021 |
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| _version_ | 1866913308744876032 |
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| author | Sardari, Naser Talebizadeh Orloski, Bryce Joseph |
| author_facet | Sardari, Naser Talebizadeh Orloski, Bryce Joseph |
| contents | Given a compact subset $Σ\subset \mathbb{R}$ (or $\mathbb{C}$) with logarithmic capacity greater than zero, we construct an explicit family of probability measures supported on $Σ$ such that their closure is all the possible weak limit measures of complete sets of conjugate algebraic integers lying inside $Σ$. We give an asymptotic formula for the number of algebraic integers with given degree and prescribed distribution. We exploit the algorithmic nature of our approach to give a family of upper bounds that converges to the smallest limiting trace-to-degree ratio of totally positive algebraic integers and improve the best previously known upper bound on the Schur-Siegel-Smyth trace problem to 1.8216. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2304_10021 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | A quantitative converse of Fekete's theorem Sardari, Naser Talebizadeh Orloski, Bryce Joseph Number Theory Mathematical Physics Given a compact subset $Σ\subset \mathbb{R}$ (or $\mathbb{C}$) with logarithmic capacity greater than zero, we construct an explicit family of probability measures supported on $Σ$ such that their closure is all the possible weak limit measures of complete sets of conjugate algebraic integers lying inside $Σ$. We give an asymptotic formula for the number of algebraic integers with given degree and prescribed distribution. We exploit the algorithmic nature of our approach to give a family of upper bounds that converges to the smallest limiting trace-to-degree ratio of totally positive algebraic integers and improve the best previously known upper bound on the Schur-Siegel-Smyth trace problem to 1.8216. |
| title | A quantitative converse of Fekete's theorem |
| topic | Number Theory Mathematical Physics |
| url | https://arxiv.org/abs/2304.10021 |