Farey codes and languages

Abstract

A word w is central if it has a minimal period π w such that | w | − π w + 2 is a period of w coprime with π w . Central words are in a two-letter alphabet A and play an essential role in combinatorics of Sturmian words. We study some new structural properties of the set PER of central words which are based on the existence of two basic bijections ψ and φ of A ∗ in PER , related to two different methods of generation, and two natural bijections θ (the ratio of periods) and η (the rate) of PER in the set of all positive irreducible fractions. In this paper we are mainly interested in sets of central words which are codes. In particular, for any positive integer n we consider the set Δ n of all central words w such that the period | w | − π w + 2 is not larger than n + 1 and | w | ≥ n . In a previous paper we proved that for each n , Δ n is a maximal prefix central code called the Farey code of order n since it is naturally associated with the Farey series of order n + 1 . New structural properties of Farey codes are given as well as of their pre-codes P n . In particular one has PER = ∪ n ≥ 0 Δ n . Moreover, for each n two languages of central words L n and M n are introduced. The language L n (resp., M n ) is called the Farey (resp., dual Farey) language of order n . The name is motivated by the fact that L n and M n give faithful representations of the set of Farey’s fractions of order n . Finally, two total orderings of PER are naturally defined in terms of maps θ and η . The notion of order of a central word relative to a language of central words is given and some general results are proved. In the case of Farey’s languages one has that the Riemann hypothesis on the Zeta function can be restated in terms of a combinatorial property of these languages.