@phdthesis{Siegel2024,
  author    = {{Maxwell Charles Siegel}},
  title     = {{{p, q}-adic Analysis and the Collatz Conjecture}},
  year      = {{2024}},
  url       = {{https://arxiv.org/abs/2412.02902}},
  abstract  = {{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 : Z_p → Z_q for an appropriate choice of distinct primes p, q with the property that
x ∈ Z\ {0} is a periodic point of H if and only if there is a p-adic integer z ∈ (Q ∩ Z_p) \ {0, 1, 2, . . .}
so that χH (z) = 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\ {0}
a periodic point of H” is essentially equivalent to “is the span of the translates of the Fourier
transform of χH (z) − 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 Z^d for any d ≥ 1.}},
}