Uniform distribution of the weighted sum-of-digits functions

Abstract

The higher-dimensional generalization of the weighted q-adic sum-of-digits functions s_{q,γ}(n), n = 0, 1, 2,... , covers several important cases of sequences investigated in the theory of uniformly distributed sequences, e.g., d-dimensional van der Corput-Halton or d-dimensional Kronecker sequences. We prove a necessary and sufficient condition for the higher-dimensional weighted q-adic sum-of-digits functions to be uniformly distributed modulo one in terms of a trigonometric product. As applications of our condition we prove some upper estimates of the extreme discrepancies of such sequences, and that the existence of distribution function g(x) = x implies the uniform distribution modulo one of the weighted q-adic sum-of-digits function s_{q,γ}(n), n = 0, 1, 2,... We also prove the uniform distribution modulo one of related sequences h_1*s_{q,γ}(n) + h_2*s_{q,γ}(n + 1), where h_1 and h_2 are integers such that h1 + h2 does not vanish and that the akin two-dimensional sequence (s_{q,γ}(n), s_{q,γ}(n + 1)) cannot be uniformly distributed modulo one if q ≥ 3. The properties of the two-dimensional sequence (s_{q,γ}(n), s_{q,γ}(n + 1)) , n = 0, 1, 2,... , will be instrumental in the proofs of the final section, where we show how the growth properties of the sequence of weights influence the distribution of values of the weighted sum-of-digits function which in turn imply a new property of the van der Corput sequence.