Numeration systems, linear recurrences, and regular sets

Abstract

A numeration system based on a strictly increasing sequence of positive integers u ₀=1, u ₁ u ₂,... expresses a non-negative integer n as a sum n=∑ ⁱ_j=0 a_ju_j. In this case we say the string a _i a _i −1 ...a₁ a₀ 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 u ₀=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=(u _j ) _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.

Supported in part by a grant from NSERC.