(q, δ)-Numeration Systems with Missing Digits

Abstract.

We consider the (q, δ) numeration system, with basis q≥2 and the set of digits {δ, δ+1, … , q+δ−1} where −(q−1)≤δ≤0. We study properties of numbers where some digits do not occur. This is analogous to the Cantor set {0.a ₁ a ₂⋯∣a _i ∈{0,2}}. We compute an asymptotic equivalent of the nth moment of the “Cantor (q, D)-distribution” which can be described as the numbers 0. w ₁ w ₂… with w _i ∈D⊆{δ, … , q+δ−1}, and each such letter can occur with the same probability 1/Card D. Furthermore, we consider n random strings according to the distribution and the expected minimum of them. We find a recursion which we solve asymptotically.