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

TL;DR

This paper establishes a small cap decoupling theorem for the parabola over non-Archimedean local fields, providing explicit constants and analyzing dependence on scale and residue prime.

Contribution

It introduces a novel decoupling theorem over non-Archimedean fields with explicit constants and detailed dependence on parameters.

Findings

Polylogarithmic dependence on scale parameter R

Polynomial dependence on residue prime (except prime 2)

Explicit constants in decoupling estimates

Abstract

We prove a small cap decoupling theorem for the parabola over a general non-Archimedean local field for which $2\neq 0$. 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.