Constructions of normal numbers with infinitely many digits

Abstract

Let $L=(L_d{)}_{d\in \mathbb{N}}$ be any ordered probability sequence, i.e., satisfying $0<L_{d+1}\le L_d$ for each $d\in \mathbb{N}$ and ${\sum }_{d\in \mathbb{N}}{L}_d=1$. We construct sequences $A=(a_i{)}_{i\in \mathbb{N}}$ on the countably infinite alphabet $\mathbb{N}$ in which each possible block of digits ${\alpha }_1,\dots ,{\alpha }_k\in \mathbb{N}$, $k\in \mathbb{N}$, occurs with frequency ${\prod }_{d=1}^k{L}_{{\alpha }_d}$. In other words, we construct $L$-normal sequences. These sequences can then be projected to normal numbers in various affine number systems, such as real numbers $x\in [0,1]$ that are normal in GLS number systems that correspond to the sequence $L$ or higher dimensional variants. In particular, this construction provides a family of numbers that have a normal Lüroth expansion.

21 pages, 3 figures