Uniform distribution of the weighted sum-of-digits functions

Kód výsledku v 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>

Výsledek na webu

<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>

Jazyk výsledku

angličtina

Název v původním jazyce

Uniform distribution of the weighted sum-of-digits functions

Popis výsledku v původním jazyce

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.

Název v anglickém jazyce

Uniform distribution of the weighted sum-of-digits functions

Popis výsledku anglicky

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.

Druh

J_ost - Ostatní články v recenzovaných periodicích

CEP obor

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OECD FORD obor

10101 - Pure mathematics

Projekt

—

Návaznosti

I - Institucionalni podpora na dlouhodoby koncepcni rozvoj vyzkumne organizace

Rok uplatnění

2021

Kód důvěrnosti údajů

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

Název periodika

Uniform Distribution Theory

ISSN

1336-913X

e-ISSN

—

Svazek periodika

16

Číslo periodika v rámci svazku

1

Stát vydavatele periodika

SK - Slovenská republika

Počet stran výsledku

34

Strana od-do

93-126

Kód UT WoS článku

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EID výsledku v databázi Scopus

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