Numeration systems, linear recurrences, and regular sets

Abstract

A numeration system based on a strictly increasing sequence of positive integers u0 = 1, u1, u2,… expresses a non-negative integer n as a sum n = Σij=0 ajuj. In this case we say the string aiai-1…a1a0 is a representation for n. If the lexicographic ordering on the representations is the same as the usual ordering of the integers, we say the numeration system is orderpreserving. In particular, if u0 = 1, then the greedy representation, obtained via the greedy algorithm, is order-preserving. We prove that, subject to some technical assumptions, if the set of all representations in an order-preserving numeration system is regular, then the sequence u = (uj)j≥0 satisfies a linear recurrence. The converse, however, is not true. The proof uses two lemmas about regular sets that may be of independent interest. The first shows that if L is regular, then the set of lexicographically greatest strings of every length in L is also regular. The second shows that the number of strings of length n in a regular language L is bounded by a constant (independent of n) iff L is the finite union of sets of the form xy*z.