Fourier Analysis of the Parity-Vector Parameterization of the Generalized Collatz px+1 maps

Abstract: Let p be an odd prime, and consider the map H_p which sends an integer x to either x/2 or (px+1)/2 depending on whether x is even or odd. The values at x=0 of arbitrary composition sequences of the maps x/2 and (px+1)/2 can be parameterized over the 2-adic integers (Z2) leading to a continuous function from Z2 to Zp which the author calls the "characteristic function" (or "numen") of H_p. Lipschitz-type estimates are given for the characteristic function when p-1 is a power of 2 and 2 is a primitive root mod p, and it is shown that the set of periodic points of H_p is equal to the set of (rational) integer values attained by the characteristic function over Z2. Additionally, although the pre-image of R under the characteristic function has zero Haar measure in the Z2, by pre-composing the characteristic function with an appropriately selected self-embedding of Z2, one can perform Fourier analysis of the aforementioned composite. Using this approach, explicit upper bounds are computed for the absolute value of a periodic point of H_p whose parity vector contains at least ceil(ln(p)/ln2)-1 zeroes between any two consecutive ones