The distribution of generalized sum-of-digits functions in residue classes

[摘要] Let Q = {Q(j)}(j=0)(infinity) be a strictly increasing sequence of integers with Q(0) = 1 and such that each Q(j) is a divisor of Q(j+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 = Sigma(j greater than or equal to 0) a(j)(n) Q(j) with digits a(j)(n) satisfying 0 less than or equal to a(j)(n) < Q(j+1)/Q(j). A Q-additive function is a function f: N --> C of the form f(n) = Sigma(j greater than or equal to 0) f(j)(a(j)(n)) where n = Sigma(j greater than or equal to 0) a(j)(n) Q(j) is the base-Q representation of n and the component functions f(j) are defined on {0, 1, ..., Q(j+1)/Q(j)-1} and satisfy f(j)(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 module 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 module any prime m. (C) 1999 Academic Press.

[发布日期] 1999-12-01