Constructions of normal numbers with infinitely many digits

TL;DR

This paper constructs sequences over infinite alphabets with prescribed digit block frequencies, leading to the creation of normal numbers in various number systems, including L"uroth expansions, expanding the class of known normal numbers.

Contribution

It introduces a method to construct $L$-normal sequences with specified digit frequencies over infinite alphabets, enabling the generation of normal numbers in diverse number systems.

Findings

Constructed sequences with prescribed digit frequencies.

Generated normal numbers in L"uroth and GLS number systems.

Provided a framework for creating normal numbers in various bases.

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 $\alpha_1, \ldots, \alpha_k \in \mathbb N$, $k \in \mathbb N$, 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 L\"uroth expansion.