The Distribution of Generalized Sum-of-Digits Functions in Residue Classes

Abstract

Let Q={Qj}∞j=0 be a strictly increasing sequence of integers with Q0=1 and such that each Qj is a divisor of Qj+1. The sequence Q is a numeration system in the sense that every positive integer n has a unique “base-Q” representation of the form n=∑j⩾0aj(n)Qj with “digits” aj(n) satisfying 0⩽aj(n)<Qj+1/Qj. A Q-additive function is a function f:N→C of the form f(n)=∑j⩾0fj(aj(n)) where n=∑j⩾0aj(n)Qj is the base-Q representation of n and the component functions fj are defined on {0, 1, …, Qj+1/Qj−1} and satisfy fj(0)=0. We study the distribution of integer-valued Q-additive functions in residue classes. Our main result gives necessary and sufficient conditions for f to be uniformly (resp. non-uniformly) distributed modulo m, for any given prime m. We apply this result to many cases, showing, for example, that the sum-of-digits functions associated with base-Q representations are uniformly distributed modulo any prime m.