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The higher-dimensional generalization of the weighted <italic>q</italic>-adic sum-of-digits functions <italic>sq,γ</italic>(<italic>n</italic>), <italic>n</italic> =0, 1, 2,..., covers several important cases of sequences investigated in the theory of uniformly distributed sequences, e.g., <italic>d</italic>-dimensional van der Corput-Halton or <italic>d</italic>-dimensional Kronecker sequences. We prove a necessary and sufficient condition for the higher-dimensional weighted <italic>q</italic>-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 <italic>g</italic>(<italic>x</italic>)= <italic>x</italic> implies the uniform distribution modulo one of the weighted <italic>q</italic>-adic sum-of-digits function <italic>sq,γ</italic> (<italic>n</italic>), <italic>n</italic> = 0, 1, 2,... We also prove the uniform distribution modulo one of related sequences <italic>h</italic>1<italic>sq, γ</italic> (<italic>n</italic>)+<italic>h</italic>2<italic>sq,γ</italic> (<italic>n</italic> +1), where <italic>h</italic>1 and <italic>h</italic>2 are integers such that <italic>h</italic>1 + <italic>h</italic>2 ≠ 0 and that the akin two-dimensional sequence <italic>sq,γ</italic> (<italic>n</italic>), <italic>sq,γ</italic> (<italic>n</italic> +1) cannot be uniformly distributed modulo one if <italic>q ≥</italic> 3. The properties of the two-dimensional sequence <italic>sq,γ</italic> (<italic>n</italic>),s<italic>q,γ</italic> (<italic>n</italic> +1), <italic>n</italic> =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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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.

Uniform Distribution of the Weighted Sum-of-Digits Functions

Department of Mathematics, Faculty of Sciences

Department of Mathematics and Informatics

Faculty of Economics of J. Selye University

Uniform distribution theory

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

{"article-title":"Uniform Distribution of the Weighted Sum-of-Digits Functions"}

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...

Uniform Distribution of the Weighted Sum-of-Digits Functions

Published Online: Oct 30, 2021

Page range: 93 - 126

Received: Feb 11, 2020

Accepted: Aug 02, 2021

DOI: https://doi.org/10.2478/udt-2021-0005

Keywords
uniform distribution, van der Corput sequence, higher-dimensional weighted sum-of-digits function, discrepancy, distribution function

© 2021 Ladislav Mišík et al., published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

Uniform Distribution of the Weighted Sum-of-Digits Functions

Published Online: Oct 30, 2021

Page range: 93 - 126

Received: Feb 11, 2020

Accepted: Aug 02, 2021

DOI: https://doi.org/10.2478/udt-2021-0005

Keywordsuniform distribution, van der Corput sequence, higher-dimensional weighted sum-of-digits function, discrepancy, distribution function

© 2021 Ladislav Mišík et al., published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

Keywords
uniform distribution, van der Corput sequence, higher-dimensional weighted sum-of-digits function, discrepancy, distribution function