On minimal expansions in redundant number systems: Algorithms and Quantitative analysis

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.