Polynomial progressions in topological fields

Let P₁, … , Pₘ ∈ K[y] be polynomials with distinct degrees, no constant terms and coefficients in a general local field K . We give a quantitative count of the number of polynomial progressions x, x+P₁(y), … , x + Pₘ(y) lying in a set S⊆ K of positive density. The proof relies on a general L∞ 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.