The exact order of discrepancy for Levin’s normal number in base 2

Hofer, Roswitha; Larcher, Gerhard

doi:10.5802/jtnb.1271

¹Institute of Financial Mathematics and Applied Number Theory, Johannes Kepler University Linz Altenbergerstraße 69, 4040 Linz, Austria

Journal de théorie des nombres de Bordeaux, Tome 35 (2023) no. 3, pp. 999-1023

Abstract (VO)
Résumé

Mordechay B. Levin in [4] has constructed a number $α$ which is normal in base 2, and such that the sequence ${2^{n} α}_{n = 0, 1, 2, ...}$ has very small discrepancy $D_{N}$ . Indeed we have $N \cdot D_{N} = 𝒪 ({(log N)}^{2})$ . That means, that $α$ is normal of extremely high quality. In this paper we show that this estimate is best possible, i.e., $N \cdot D_{N} \geq c \cdot {(log N)}^{2}$ for infinitely many $N$ .

Dans [4], Mordechay B. Levin a construit un nombre $α$ qui est normal en base 2 et tel que la suite ${2^{n} α}_{n = 0, 1, 2, ...}$ a une très faible discrépance $D_{N}$ . En effet, nous avons $N \cdot D_{N} = 𝒪 ({(log N)}^{2})$ . Cela signifie que $α$ est un nombre normal de très haute qualité. Dans cet article, nous montrons que cette estimation est la meilleure possible, c’est-à-dire que $N \cdot D_{N} \geq c \cdot {(log N)}^{2}$ pour une infinité de $N$ .

Reçu le : 2022-09-19
Révisé le : 2022-12-13
Accepté le : 2022-12-14
Publié le : 2024-03-04

DOI : 10.5802/jtnb.1271

Classification : 11K16, 11K38
Keywords: normal numbers, Levin’s number, uniform distribution of sequences, discrepancy

Affiliations des auteurs :

Hofer, Roswitha ¹ ; Larcher, Gerhard ¹

¹ Institute of Financial Mathematics and Applied Number Theory, Johannes Kepler University Linz Altenbergerstraße 69, 4040 Linz, Austria

Licence :

CC-BY-ND 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     title = {The exact order of discrepancy for {Levin{\textquoteright}s} normal number in base 2},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     pages = {999--1023},
     year = {2023},
     publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
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Hofer, Roswitha; Larcher, Gerhard. The exact order of discrepancy for Levin’s normal number in base 2. Journal de théorie des nombres de Bordeaux, Tome 35 (2023) no. 3, pp. 999-1023. doi: 10.5802/jtnb.1271

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