Uniform Distribution of the Weighted Sum-of-Digits Functions

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 h₁s_{q, γ} (n)+h₂s_q,γ (n +1), where h₁ and h₂ are integers such that h₁ + h₂ ≠ 0 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.