On uniform polynomial approximation
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arXiv
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866916243241435136 |
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| author | Poëls, Anthony |
| author_facet | Poëls, Anthony |
| contents | Let $n$ be a positive integer and $ξ$ a transcendental real number. We are interested in bounding from above the uniform exponent of polynomial approximation $\widehatω_n(ξ)$. Davenport and Schmidt's original 1969 inequality $\widehatω_n(ξ)\leq 2n-1$ was improved recently, and the best upper bound known to date is $2n-2$ for each $n\geq 10$. In this paper, we develop new techniques leading us to the improved upper bound $2n-\frac{1}{3}n^{1/3}+\mathcal{O}(1)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_07219 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On uniform polynomial approximation Poëls, Anthony Number Theory 11J13(Primary), 11J82 (Secondary) Let $n$ be a positive integer and $ξ$ a transcendental real number. We are interested in bounding from above the uniform exponent of polynomial approximation $\widehatω_n(ξ)$. Davenport and Schmidt's original 1969 inequality $\widehatω_n(ξ)\leq 2n-1$ was improved recently, and the best upper bound known to date is $2n-2$ for each $n\geq 10$. In this paper, we develop new techniques leading us to the improved upper bound $2n-\frac{1}{3}n^{1/3}+\mathcal{O}(1)$. |
| title | On uniform polynomial approximation |
| topic | Number Theory 11J13(Primary), 11J82 (Secondary) |
| url | https://arxiv.org/abs/2405.07219 |