On Minimal Expansions in Redundant Number Systems: Algorithms and Quantitative Analysis

Abstract

We consider digit expansions in base q≥2 with arbitrary integer digits such that the length of the expansion plus the sum of the absolute values of the digits is minimal. Since this does not determine a unique minimal representation, we describe some reduced minimal expansions.

We completely characterize its syntactical properties, give a simple algorithm to compute the reduced minimal expansion and a formula to compute a single digit without having to compute the others, and we calculate the average cost of such an expansion.