Uniform distribution of the weighted sum-of-digits functions

Result code in IS VaVaI

<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F67985807%3A_____%2F21%3A00546811" target="_blank" >RIV/67985807:_____/21:00546811 - isvavai.cz</a>

Result on the web

<a href="http://dx.doi.org/10.2478/udt-2021%E2%80%930005" target="_blank" >http://dx.doi.org/10.2478/udt-2021%E2%80%930005</a>

DOI - Digital Object Identifier

<a href="http://dx.doi.org/10.2478/udt-2021-0005" target="_blank" >10.2478/udt-2021-0005</a>

Result language

angličtina

Original language name

Uniform distribution of the weighted sum-of-digits functions

Original language description

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.

Czech name

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Czech description

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Type

J_ost - Miscellaneous article in a specialist periodical

CEP classification

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OECD FORD branch

10101 - Pure mathematics

Project

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Continuities

I - Institucionalni podpora na dlouhodoby koncepcni rozvoj vyzkumne organizace

Publication year

2021

Confidentiality

S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

Name of the periodical

Uniform Distribution Theory

ISSN

1336-913X

e-ISSN

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Volume of the periodical

16

Issue of the periodical within the volume

1

Country of publishing house

SK - SLOVAKIA

Number of pages

34

Pages from-to

93-126

UT code for WoS article

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EID of the result in the Scopus database

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