Geodesic
Polynomial progressions in topological fields
Forum of Mathematics, Sigma, Tome 12 (2024)

Voir la notice de l'article provenant de la source Cambridge University Press

Let $P_1, \ldots , P_m \in \mathbb {K}[\mathrm {y}]$ be polynomials with distinct degrees, no constant terms and coefficients in a general local field $\mathbb {K}$. 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 $S\subseteq \mathbb {K}$ 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. 
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     author = {Ben Krause and Mariusz Mirek and Sarah Peluse and James Wright},
     title = {Polynomial progressions in topological fields},
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Ben Krause; Mariusz Mirek; Sarah Peluse; James Wright. Polynomial progressions in topological fields. Forum of Mathematics, Sigma, Tome 12 (2024). doi: 10.1017/fms.2024.104

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