Constructions of normal numbers with infinite digit sets

Abstract

Let $L={\left(L_d\right)}_{d\in N}$ be any ordered probability sequence, i.e., satisfying $0<L_{d+1}\le L_d$ for each $d\in N$ and ${\sum }_{d\in N}{L}_d=1$. We construct sequences $A={\left(a_i\right)}_{i\in N}$ on the countably infinite alphabet $N$ in which each possible block of digits ${\alpha }_1,\dots ,{\alpha }_k\in N$, $k\in N$, 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.