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 of n = Zij=0 ajuj. In this case we say the string aiai-1 . . . a1a0 is a representation for n. If ged (u0,u1, . . . ) = g, then every sufficiently large multiple of g has some representation.

If the lexicographic ordering on the representations is the same as the usual ordering of the integers, we say the numeration system is order-preserving. 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 is 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.