The Collatz Conjecture & Non-Archimedean Spectral Theory -- Part II -- $\left(p,q\right)$-Adic Fourier Analysis and Wiener's Tauberian Theorem

Abstract

This paper gives an overview of $\left(p,q\right)$-adic Fourier theory - the Fourier theory of functions from the $p$-adic numbers to the $q$-adic numbers, where $p$ and $q$ are distinct primes - which we then use to prove a novel $\left(p,q\right)$-adic generalization of Norbert Wiener's celebrated Tauberian Theorem. Letting $K$ be a metrically complete, algebraically closed local field of residue characteristic $q$, letting $C\left(\mathbb{Z}_{p},K\right)$ be the Banach space of continuous functions $\mathbb{Z}_{p}\rightarrow K$, and letting $d\mu$ be a $\left(p,q\right)$-adic measure (a continuous linear functional $C\left(\mathbb{Z}_{p},K\right)\rightarrow K)$, the $\left(p,q\right)$-adic Wiener Tauberian Theorem (WTT) we prove establishes the equivalence of the density of the span of translates of $d\mu$'s Fourier-Stieltjes Transform and the non-vanishing of the Radon-Nikodym derivative of $d\mu$ at all points in $\mathbb{Z}_{p}$ where the derivative exists in $K$.