On simply normal numbers with digit dependencies

Verónica Becher; Agustín Marchionna; Gérald Tenenbaum

paper

On simply normal numbers with digit dependencies

arXiv:2304.06850

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, $\sum_{p \neq \in P} 1/ p = \infty$ . We then provide an effective bound for the discrepancy.