Benford's law for the 3x + 1 function

Abstract

Benford's law (to base B) for an infinite sequence {xk : k ≥ 1} of positive quantities xk, is the assertion that {log Bxk : k ≥ 1} is uniformly distributed (mod 1). The 3x + 1 function T(n) is given by T(n) = (3n + 1)/2 if n is odd, and T(n) = n/2 if n is even. This paper studies the initial iterates xk = T (k)(x0) for 1 ≤ k ≤ N of the 3x + 1 function, where N is fixed. It shows that for most initial values x0, such sequences approximately satisfy Benford's law, in the sense that the discrepancy of the finite sequence {logB xk : 1 ≤ k ≤ N} is small. © 2006 London Mathematical Society.