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

2022/08/18 by Maxwell C. Siegel, Siegel, Maxwell C.
Mathematics · #advanced mathematical theories

paper · pdf · doi:10.48550/arxiv.2208.11082

Abstract

This paper gives an overview of (p,q)-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 (p,q)-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(ℤ_p,K) be the Banach space of continuous functions ℤ_p→ K, and letting dμ be a (p,q)-adic measure (a continuous linear functional C(ℤ_p,K)→ K), the (p,q)-adic Wiener Tauberian Theorem (WTT) we prove establishes the equivalence of the density of the span of translates of dμ's Fourier-Stieltjes Transform and the non-vanishing of the Radon-Nikodym derivative of dμ at all points in ℤ_p where the derivative exists in K.

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• (p,q)-adic Analysis and the Collatz Conjecture
• The Collatz Conjecture & Non-Archimedean Spectral Theory: Part I -- Arithmetic Dynamical Systems and Non-Archimedean Value Distribution Theory
• The Collatz Conjecture & Non-Archimedean Spectral Theory -- Part I.5 -- How To Write The (Weak) Collatz Conjecture As A Contour Integral
• Fourier transforms and p-adic "Weil II"
• Another look at p-adic Fourier-theory