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Abstract A nonlinear version of Roth's theorem states that dense sets of integers contain configurations of the form , , . We obtain a multidimensional version of this result, which can be regarded as a first step toward effectivising those cases of the multidimensional polynomial Szemerédi theorem involving polynomials with distinct degrees. In addition, we prove an effective “popular” version of this result, showing that every dense set has some non‐zero such that the number of configurations with difference parameter is almost optimal. Perhaps surprisingly, the quantitative dependence in this result is exponential, compared to the tower‐type bounds encountered in the popular linear Roth theorem.more » « lessFree, publicly-accessible full text available November 1, 2025
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Abstract 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 andp-adic analysis.more » « less
