Numeration systems defined by addition rules

Abstract. We investigate the problem of reconstructing polynomial-based ternary number systems when the expansion of certain elements — the pairwise sums of digits — is given. We call these expansions the “addition rules” for the system. We define a randomized process for expanding arbitrary elements based on these addition rules. For some rules, the expansion is not-well defined. We search for addition rules below a certain length that yield well-defined expansion for all elements, which we call the uniqueness property. We also give algorithms for the reconstruction of the base polynomial and the digits of the numeration system from the addition rules. Finally, we formulate some open problems for further research.