Constructions of normal numbers with infinite digit sets

Let L=(L_d)_d∈N be any ordered probability sequence, i.e., satisfying 0<L_d+1≤L_d for each d∈N and ∑_d∈NL_d=1. We construct sequences A=(a_i)_i∈N on the countably infinite alphabet N in which each possible block of digits α₁,…,α_k∈N, k∈N, occurs with frequency ∏_d=1^kL_αd. 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 Lüroth expansion.