High-low analysis and small cap decoupling over non-Archimedean local fields

Abstract

We prove a small cap decoupling theorem for the parabola over a general non-Archimedean local field for which $2\ne 0$. We obtain polylogarithmic dependence on the scale parameter $R$ and polynomial dependence in the residue prime, except for the prime 2 for which the polynomial depends on degree. Our constants are fully explicit.

34 pages New version: corrects various minor errors, slightly improves numerology. Three applications are added to real-variable integrals