Polynomial progressions in topological fields

Abstract

Let $P_1,\dots ,P_m\in K[y]$ 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 $x,x+P_1(y),\dots ,x+P_m(y)$ 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.

51 pages, no figures, suggestions from the referees reports incorporated. Accepted for publication in the Forum of Mathematics, Sigma

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• Arithmetic progressions in sets of fractional dimension
• On the polynomial Szemerédi theorem in finite fields
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• Bounds for sets with no polynomial progressions
• The Kelley--Meka bounds for sets free of three-term arithmetic progressions
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Cited by in corpus (1)

• Pointwise convergence of bilinear polynomial averages over the primes