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Polynomial Progressions in Topological Fields by Krause, Ben; Mirek, Mariusz; Peluse, Sarah; Wright, James is a scholarly article available to read on EtoBox.

What is Polynomial Progressions in Topological Fields about?

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.

Author

Krause, Ben; Mirek, Mariusz; Peluse, Sarah; Wright, James

Published

2022

Language

EN