Restricted projections and Fourier decoupling in $\mathbb{Q}_p^n$

TL;DR

This paper establishes a restricted projection theorem for Borel sets in p-adic spaces, generalizing real results, and provides explicit Fourier decoupling theorems with concrete constants.

Contribution

It introduces a p-adic restricted projection theorem and an explicit Fourier decoupling theorem, extending prior real-space results to the p-adic setting.

Findings

Proved a restricted projection theorem for Borel sets in $Q_p^n$ for $p>n$.

Derived explicit constants for covering numbers in the p-adic context.

Established a fully explicit Fourier decoupling theorem for the p-adic moment curve.

Abstract

We prove a restricted projection theorem for Borel subsets of $\mathbb{Q}_p^n$ in the regime $p>n$. This generalizes results of Gan-Guo-Wang in the real setting. Our result is effective in the sense that explicit constants are obtained for various covering numbers. Along the way, we prove a fully explicit Fourier decoupling theorem for the moment curve in $p$-adic Cartesian space.