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 Trudy Mat. Inst. Steklova, 2021, Volume 314, Pages 97–102 (Mi tm4187)

On Irregularity of Finite Sequences

S. V. Konyagin

Steklov Mathematical Institute of Russian Academy of Sciences, ul. Gubkina 8, Moscow, 119991 Russia

Abstract: A sequence $(x_1,x_2,…,x_{N+d})$ of numbers in $[0,1)$ is said to be $N$-regular with at most $d$ irregularities if for every $n=1,…,N$ each of the intervals $[0,1),[1,2),…,[n-1,n)$ contains at least one element of the sequence $(nx_1,nx_2,…,nx_{n+d})$. The maximum $N$ for which there exists an $N$-regular sequence with at most $d$ irregularities is denoted by $s(d)$. We show that $s(d)\ge 2d$ for any positive integer $d$ and that $s(d)<200d$ for all sufficiently large $d$.

DOI: https://doi.org/10.4213/tm4187

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English version:
Proceedings of the Steklov Institute of Mathematics, 2021, 314, 90–95

Bibliographic databases:

UDC: 511.216
Received: August 31, 2020
Revised: January 20, 2021
Accepted: February 26, 2021

Citation: S. V. Konyagin, “On Irregularity of Finite Sequences”, Analytic and Combinatorial Number Theory, Collected papers. In commemoration of the 130th birth anniversary of Academician Ivan Matveevich Vinogradov, Trudy Mat. Inst. Steklova, 314, Steklov Math. Inst., Moscow, 2021, 97–102; Proc. Steklov Inst. Math., 314 (2021), 90–95

Citation in format AMSBIB
\Bibitem{Kon21} \by S.~V.~Konyagin \paper On Irregularity of Finite Sequences \inbook Analytic and Combinatorial Number Theory \bookinfo Collected papers. In commemoration of the 130th birth anniversary of Academician Ivan Matveevich Vinogradov \serial Trudy Mat. Inst. Steklova \yr 2021 \vol 314 \pages 97--102 \publ Steklov Math. Inst. \publaddr Moscow \mathnet{http://mi.mathnet.ru/tm4187} \crossref{https://doi.org/10.4213/tm4187} \transl \jour Proc. Steklov Inst. Math. \yr 2021 \vol 314 \pages 90--95 \crossref{https://doi.org/10.1134/S0081543821040052} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000705530700005} \scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-85116550182} 

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