Direct definition of a ternary infinite square-free sequence

We propose a new ternary infinite (even full-infinite) square-free sequence. The sequence is defined both by an iterative method and by a direct definition. Both definitions are analogous to those of the Thue-Morse sequence. The direct definition is given by a deterministic finite automaton with output. In short, the sequence is automatic.

💡 Research Summary

The paper introduces a novel ternary infinite square‑free sequence that parallels the classic Thue‑Morse construction. The authors first define two permutations on the alphabet Σ = {1, 2, 3}: σ swaps 1 and 2, while ρ swaps 2 and 3. Using these, they build an operator ϕ(a) = σ(a)·a·ρ(a) which triples the length of any word a. Starting from the single symbol 2, repeated application yields words ϕⁿ(2) of length 3ⁿ. The central result (Theorem 1) states that for every n, ϕⁿ(2) is square‑free, i.e., it contains no factor of the form ww with w non‑empty.

The proof proceeds via two lemmas. Lemma 2 shows that each block of three consecutive symbols in ϕⁿ(2) is a permutation of (1, 2, 3). Lemma 3 defines an extraction operator f that picks the middle symbol of each triple; it proves f∘ϕ = ϕ∘f, implying that the middle third of ϕⁿ⁺¹(2) reproduces ϕⁿ(2). With these tools, the authors conduct a case analysis on the possible length ℓ of a hypothetical square: (i) ℓ = 1 or 2, (ii) ℓ divisible by 3, and (iii) ℓ ≥ 4 with ℓ ≠ 0 (mod 3). In each case they derive contradictions using the structure of σ‑ and ρ‑images and the induction hypothesis, thereby establishing square‑freeness for all n.

To obtain a direct, non‑iterative description, the paper adopts balanced ternary representation. Any integer i (positive, zero, or negative) is uniquely expressed as i = ∑{k=0}^{ν} u_k 3^k with digits u_k ∈ {−1, 0, +1}. Three permutations π{−1}, π_0, π_{+1} on {−1, 0, +1} are defined: π_{−1} swaps −1 and 0, π_0 is the identity, and π_{+1} swaps 0 and +1. The sequence {b_i} is then defined by applying the corresponding permutations in order of the balanced ternary digits, starting from 0: b_i = π_{u_ν} ∘ … ∘ π_{u_0}(0).
This definition can be realized by a deterministic finite automaton with output (DFAO) having three states q_{−1}, q_0, q_{+1}, each emitting the symbol −1, 0, +1 respectively. The transition diagram (Figure 1) is a simple directed graph where reading a digit −1, 0, +1 triggers the corresponding permutation. Consequently, {b_i} is 3‑automatic.

Theorem 4 proves that the finite segment {b_i}{i=−(3ⁿ−1)/2}^{(3ⁿ−1)/2} coincides with ϕⁿ(0), where ϕⁿ(0) is obtained from ϕⁿ(2) by replacing 1, 2, 3 with −1, 0, +1 respectively. Hence the full bi‑infinite sequence {b_i}{i∈ℤ}=ϕ^∞(0) inherits the square‑free property from Theorem 1. This mirrors the dual definitions of the Thue‑Morse sequence: a recursive morphism ψ and a direct binary‑digit‑sum function s₂(i). Here, ϕ plays the role of ψ, while the balanced ternary digits together with the π‑permutations play the role of s₂(i).

The authors note that unlike earlier ternary square‑free sequences (Arshon, Leech, Zech), which are semi‑infinite and often described via uniform tag sequences, their construction yields a full‑infinite, automatic sequence with an extremely simple DFAO (only three states). They suggest potential applications in information theory, random physical systems, and quasiperiodic tilings, citing the ease of generation as an advantage.

While the paper successfully introduces the sequence and establishes its key properties, some aspects could be strengthened. The case analysis for ℓ ≥ 4 is dense and could benefit from clearer tabulation of sub‑cases. A more explicit discussion of the automaton’s complexity (e.g., state‑transition matrix, eigenvalues) and of combinatorial statistics of the sequence (frequency of each symbol, balance properties) would deepen the contribution. Nonetheless, the work provides a valuable addition to the theory of automatic sequences and combinatorics on words, presenting a clean ternary analogue of the celebrated Thue‑Morse sequence.