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 $α_1, \ldots, α_k \in \mathbb N$, $k \in \mathbb N$, occurs with frequency $\prod_{d=1}^k L_{α_d}$. 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.

Figures (3)

• Figure 1: The graph of the Lüroth transformation $T_L$.
• Figure 2: The first six levels of the $L$-tree.
• Figure 3: The graph of a GLS transformation restricted to $(\ell_1, r_1] \cup (\ell_2,r_2] \cup (\ell_3,r_3]$ with $\varepsilon_1=\varepsilon_2=1$ and $\varepsilon_3=0$.

Theorems & Definitions (22)

• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Proposition 3.1
• proof
• Lemma 3.1
• Lemma 3.2
• proof
• Proposition 3.2