Benford's Law for the 3 x + 1 Function

Benford's law (to base B ) for an infinite sequence { x k : k ⩾ 1} of positive quantities x k is the assertion that {log B x k : k ⩾ 1} is uniformly distributed (mod 1). The 3 x + 1 function T ( n ) is given by T ( n ) = (3 n + 1)/2 if n is odd, and T ( n ) = n /2 if n is even. This paper studies the initial iterates x k = T ( k ) ( x 0 ) for 1 ⩽ k ⩽ N of the 3 x + 1 function, where N is fixed. It shows that for most initial values x 0 , such sequences approximately satisfy Benford's law, in the sense that the discrepancy of the finite sequence {log B x k : 1 ⩽ k ⩽ N } is small.