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 Hp 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 Hp. 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 Hp 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 Hp whose parity vector contains at least ceil(ln(p)/ln2)-1 zeroes between any two consecutive ones