Fifes Theorem for (7/3)-Powers

We prove a Fife-like characterization of the infinite binary (7/3)-power-free words, by giving a finite automaton of 15 states that encodes all such words. As a consequence, we characterize all such words that are 2-automatic.

💡 Research Summary

The paper presents a comprehensive Fife‑type characterization of infinite binary words that avoid 7/3‑powers, i.e., factors whose exponent is at least 7/3. The authors begin by recalling the notion of an α‑power and the special role of the exponent 7/3: it is the largest exponent for which the number of α‑power‑free words of length n grows only polynomially. They then introduce the Thue‑Morse morphism μ defined by μ(0)=01 and μ(1)=10, and show that every infinite 7/3‑power‑free binary word x can be uniquely expressed as a nested application of μ preceded by a finite prefix taken from a small set P={ε,0,00,1,11}. Formally,  x = p_{i₁} μ(p_{i₂} μ(p_{i₃} μ(⋯))), where each p_{i_j} belongs to P and the sequence of indices (i₁,i₂,…) is uniquely determined by the first five letters of x. This is Theorem 1 and its Corollary 2, which also handle the case where the infinite tail is either the fixed point μ^ω(0) (the Thue‑Morse word t) or its complement μ^ω(1).

Having obtained a canonical factorization, the authors turn to the problem of deciding which index sequences actually correspond to 7/3‑power‑free words. For any finite word w over the alphabet {0,1,2,3,4} they define a transformation C_w that inserts the corresponding prefixes p_{i} and applies μ repeatedly, and they consider the language  F_w = { x ∈ {0,1}^ω | C_w(x) ∈ F_{7/3} }, where F_{7/3} denotes the set of all infinite 7/3‑power‑free words. Lemma 6 shows that, despite the infinite number of possible w, only fifteen distinct non‑empty languages F_w arise. These fifteen languages are related by a system of equalities displayed in Figure 1; many of them are empty because the associated prefix already creates a forbidden 7/3‑power.

The proof of the equalities relies on three technical lemmas. Lemma 3 describes how a prefix of μ(x) lifts to a double prefix of x via μ^{-1}. Lemma 4 establishes that μ preserves both the exponent of a word and β‑power‑freeness for β≥2, while Lemma 5 shows that a long p‑periodic prefix of μ(x) with exponent ≥2 forces a corresponding (p/2)‑periodic prefix of x with the same exponent. Using these tools, the authors argue case by case that if one side of an equality yields a 7/3‑power‑free word, then so does the other, often because one word is a μ‑image of the other or shares a long prefix that cannot contain a higher exponent.

The fifteen languages correspond to the fifteen states of a finite automaton. A transition from state F_w to state F_{w’} occurs when a new prefix p_i is appended and μ is applied, exactly mirroring the definition of C_w. Consequently, any infinite binary word is 7/3‑power‑free if and only if its encoding as a sequence of indices (i₁,i₂,…) produces an infinite walk in this automaton. This automaton thus provides a decision procedure: to test whether a given infinite word is 7/3‑power‑free, one simply follows the corresponding path; if the path stays within the fifteen states forever, the word is safe.

A major corollary concerns 2‑automatic sequences, i.e., sequences generated by a finite automaton reading the binary expansion of the index. Because a 2‑automatic sequence can be described by a finite set of states and transitions, the authors can check whether the induced index sequence (i₁,i₂,…) stays inside the 15‑state automaton. They prove that a 2‑automatic binary sequence is 7/3‑power‑free precisely when its generating automaton can be synchronized with the 15‑state automaton, yielding a complete classification of 2‑automatic 7/3‑power‑free words.

In summary, the paper extends the classical Fife theorem—originally characterizing overlap‑free (2‑plus) words—to the more delicate setting of 7/3‑power avoidance. By introducing a canonical μ‑based factorization and constructing a compact 15‑state automaton, the authors achieve both a structural understanding and an algorithmic tool for recognizing such words. The work bridges combinatorics on words with automata theory and opens the door to further algorithmic investigations of power‑free languages, especially in the context of automatic sequences and related applications in coding theory and symbolic dynamics.