On the number of α-power-free binary words for 2<α≤7/3

Abstract

a b s t r a c tWe study the number u α (n) of α-power-free binary words of length n, and the asymptotics of this number when n tends to infinity, for a fixed rational number α in (2, 7/3]. For any such α, we prove a structure result that allows us to describe constructively the sequence u α (n) as a 2-regular sequence. This provides an algorithm that computes the number u α (n) in logarithmic time, for fixed α. Then, generalizing recent results on 2 + -free words, we describe the asymptotic behaviour of u α (n) in terms of joint spectral quantities of a pair of matrices that one can efficiently construct, given a rational number α.For α = 7/3, we compute the automaton and give sharp estimates for the asymptotic behaviour of u α (n).

Fife's Theorem for (7/3)-Powers

“…Remark 8. Blondel, Cassaigne, and Jungers [5] obtained a similar result, and even more general ones, for finite words. The main advantage to our construction is its simplicity.…”

“…We point out that these words are of particular interest, because 7 3 is the largest exponent α such that there are only polynomiallymany α-power-free words of length n [9]. The exponent 7 3 plays a special role in combinatorics on words, as testified to by the many papers mentioning this exponent (e.g., [10,14,9,11,1,5]).…”

The first-order theory of binary overlap-free words is decidable

“…Again, this is because the only accepting path reachable from this state consists of an infinite tail of 1's, which does not result in a 7 3 -power-free word. As an application, let us reprove a result from [11]: 4 There exist uncountably many infinite 7 3 -power-free binary words, each containing arbitrarily large overlaps.…”

Similarity density of the Thue-Morse word with overlap-free infinite binary words

“…A third avenue of investigation is to consider what occurs in words that avoid higher powers in place of being overlap-free (which are essentially (2 + ε)or 2 + -powers). In fact, there is a Fife automaton characterizing 7 3 -power-free infinite binary words having the same encoding mechanism as the Fife automaton for overlap-free infinite binary words but with more states and different transitions [3,9]. However, initial inspection of the automaton for 7 3 -power-free infinite binary words suggests that our proof of Theorem 18 cannot be extended to account for all 7 3 -power-free infinite binary words because there are many more edges labeled 2 and 4 in the Fife automaton for 7 3 -power-free infinite binary words, resulting in valid paths that contain infinitely many 2s and 4s, but our proof of Theorem 18 heavily relied on there being at most one occurrence of 2 or 4 (which must be preceeded by a string of 0s if it occurs) in the path taken through the automaton so that the infinite binary word corresponding to the path eventually "lags behind" the prefixes t n of t in the sense that each successive n th symbol in the path can only generate positions prior to 2 n .…”