2024/12/03 by Maxwell C. Siegel, Siegel, Maxwell Charles
Mathematics · #Benford’s Law and Fraud Detection
paper · pdf · doi:10.48550/arxiv.2412.02902
What use can there be for a function from the p-adic numbers to the q-adic numbers, where p and q are distinct primes? The traditional answer, courtesy of the half-century old theory of non-archimedean functional analysis: not much. It turns out this judgment was premature. '(p,q)-adic analysis' of this sort appears to be naturally suited for studying the infamous Collatz map and similar arithmetical dynamical systems. Given such a map H:ℤ→ℤ, one can construct a function χH:ℤp→ℤq for an appropriate choice of distinct primes p,q with the property that x∈ℤ\backslash\ 0\ is a periodic point of H if and only if there is a p-adic integer \mathfrakz∈(ℚ∩ℤp)\backslash\ 0,1,2,…\ so that χH(\mathfrakz)=x. By generalizing Monna-Springer integration theory and establishing a (p,q)-adic analogue of the Wiener Tauberian Theorem, one can show that the question 'is x∈ℤ\backslash\ 0\ a periodic point of H?' is essentially equivalent to 'is the span of the translates of the Fourier transform of χH(\mathfrakz)-x dense in an appropriate non-archimedean function space?' This presents an exciting new frontier in Collatz research, and these methods can be used to study Collatz-type dynamical systems on the lattice ℤd for any d≥1.