1. bookVolume 16 (2021): Issue 1 (June 2021)
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30 Dec 2013
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Uniform Distribution of the Weighted Sum-of-Digits Functions

Published Online: 30 Oct 2021
Page range: 93 - 126
Received: 11 Feb 2020
Accepted: 02 Aug 2021
Journal Details
License
Format
Journal
First Published
30 Dec 2013
Publication timeframe
1 time per year
Languages
English
Abstract

The higher-dimensional generalization of the weighted q-adic sum-of-digits functions sq,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 sq,γ (n), n = 0, 1, 2,... We also prove the uniform distribution modulo one of related sequences h1sq, γ (n)+h2sq,γ (n +1), where h1 and h2 are integers such that h1 + h2 ≠ 0 and that the akin two-dimensional sequence sq,γ (n), sq,γ (n +1) cannot be uniformly distributed modulo one if q ≥ 3. The properties of the two-dimensional sequence sq,γ (n),sq,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.

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MSC 2010

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