Polynomial progressions in topological fields
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arXiv
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| Auteurs principaux: | , , , |
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| Format: | Preprint |
| Publié: |
2022
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| _version_ | 1866915034223869952 |
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| author | Krause, Ben Mirek, Mariusz Peluse, Sarah Wright, James |
| author_facet | Krause, Ben Mirek, Mariusz Peluse, Sarah Wright, James |
| contents | Let $P_1, \ldots, P_m \in K[y]$ be polynomials with distinct degrees, no constant terms and coefficients in a general locally compact topological field $K$. We give a quantitative count of the number of polynomial progressions $x, x+P_1(y), \ldots, x + P_m(y)$ lying in a set $S\subseteq K$ of positive density. The proof relies on a general $L^{\infty}$ inverse theorem which is of independent interest. This inverse theorem implies a Sobolev improving estimate for multilinear polynomial averaging operators which in turn implies our quantitative estimate for polynomial progressions. This general Sobolev inequality has the potential to be applied in a number of problems in real, complex and $p$-adic analysis. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2210_00670 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Polynomial progressions in topological fields Krause, Ben Mirek, Mariusz Peluse, Sarah Wright, James Number Theory Classical Analysis and ODEs Combinatorics Let $P_1, \ldots, P_m \in K[y]$ be polynomials with distinct degrees, no constant terms and coefficients in a general locally compact topological field $K$. We give a quantitative count of the number of polynomial progressions $x, x+P_1(y), \ldots, x + P_m(y)$ lying in a set $S\subseteq K$ of positive density. The proof relies on a general $L^{\infty}$ inverse theorem which is of independent interest. This inverse theorem implies a Sobolev improving estimate for multilinear polynomial averaging operators which in turn implies our quantitative estimate for polynomial progressions. This general Sobolev inequality has the potential to be applied in a number of problems in real, complex and $p$-adic analysis. |
| title | Polynomial progressions in topological fields |
| topic | Number Theory Classical Analysis and ODEs Combinatorics |
| url | https://arxiv.org/abs/2210.00670 |