On simply normal numbers with digit dependencies

IF 0.8 3区数学 Q2 MATHEMATICS

Mathematika Pub Date : 2023-07-14 DOI:10.1112/mtk.12216

Verónica Becher, Agustín Marchionna, Gérald Tenenbaum

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引用次数: 0

Abstract

Given an integer $b ⩾ 2$ and a set $P$ of prime numbers, the set $T_{P}$ of Toeplitz numbers comprises all elements of [0, b[ whose digits ${(a_{n})}_{n ⩾ 1}$ in the base-b expansion satisfy $a_{n} = a_{p n}$ for all $p \in P$ and $n ⩾ 1$ . Using a completely additive arithmetical function, we construct a number in $T_{P}$ that is simply Borel normal if, and only if, $\sum_{p \notin P} 1 / p = \infty$ . We then provide an effective bound for the discrepancy.

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关于具有数字依赖性的简单正规数

给定整数$b\geqslant 2$和素数的集合$P$，Toeplitz数的集合$T_P$包括$[0，b$$的所有元素，其数字$（a_n）_｛n\geqslant 1｝$在基-$b$展开中对于P$中的所有$P\和$n\geqsant 1｝满足$a_n=a_｛pn｝$1/p=\infty$。然后，我们为差异提供了一个有效的界限。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Mathematika MATHEMATICS, APPLIED-MATHEMATICS

CiteScore

1.40

自引率

0.00%

发文量

审稿时长

>12 weeks

期刊介绍： Mathematika publishes both pure and applied mathematical articles and has done so continuously since its founding by Harold Davenport in the 1950s. The traditional emphasis has been towards the purer side of mathematics but applied mathematics and articles addressing both aspects are equally welcome. The journal is published by the London Mathematical Society, on behalf of its owner University College London, and will continue to publish research papers of the highest mathematical quality.