Constructions of normal numbers with infinitely many digits

Let L=(Ld)d ∈ N be any ordered probability sequence, i.e., satisfying 0 < Ld+1 ≤ Ld for each d ∈ N and ∑d ∈ N Ld =1. We construct sequences A = (aᵢ)i ∈ N on the countably infinite alphabet N in which each possible block of digits α₁, …, αₖ ∈ N, k ∈ N, occurs with frequency ∏d=1ᵏ Lαd. 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 ∈ [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.