Constructions of normal numbers with infinitely many digits

Let L= (Ld) ₃ ₍ be any ordered probability sequence, i. e. , satisfying 0 < L₃+₁ Ld for each d N and ₃ ₍ Ld =1. We construct sequences A = (aᵢ) ₈ ₍ on the countably infinite alphabet N in which each possible block of digits ₁, , ₖ N, k N, occurs with frequency ₃=₁ᵏ L₃. 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.