Basic digit sets for radix representation of the integers

Abstract

Let Z denote the set of integers. A digit set D ⊂ Z is basic for base β ∊ Z if the set of polynomials {dmβ^m + dm−1 + … + d1 β+d0 | dI ∊ D} contains a unique representation for every n ε Z. We give necessary and sufficient conditions for D to be basic for β. We exhibit efficient procedures for verifying that D is basic for β, and for computing the representation of any n ε Z when a representation exists. There exist D, & with D basic for β where max {|d| | d ∊ D} > |β|, and more generally, an infinite class of basic digit sets is shown to exist for every base β with |β| ≥ 3. The natural extension to infinite precision radix representation using basic digit sets is considered and a summary of results is presented.