Fewest repetitions in infinite binary words

A square is the concatenation of a nonempty word with itself. A word has period p if its letters at distance p match. The exponent of a nonempty word is the quotient of its length over its smallest period. In this article we give a proof of the fact that there exists an infinite binary word which contains finitely many squares and simultaneously avoids words of exponent larger than 7/3. Our infinite word contains 12 squares, which is the smallest possible number of squares to get the property, and 2 factors of exponent 7/3. These are the only factors of exponent larger than 2. The value 7/3 introduces what we call the finite-repetition threshold of the binary alphabet. We conjecture it is 7/4 for the ternary alphabet, like its repetitive threshold.

💡 Research Summary

The paper investigates a refined version of the classical repetition threshold problem by asking how few repetitions (squares) an infinite binary word can contain while simultaneously avoiding any factor whose exponent exceeds a given bound. The authors introduce the notion of a finite‑repetition threshold (FRT): the smallest real number α such that there exists an infinite word over a fixed alphabet that contains only finitely many squares and has no factor of exponent larger than α.

For the binary alphabet, they prove that the FRT equals 7/3 (≈2.333…). They construct an explicit infinite binary word that satisfies two stringent conditions: (i) it contains exactly twelve distinct squares and no other squares, and (ii) the only factors whose exponent exceeds 2 are two occurrences of exponent 7/3, namely the words 001001 and 110110. These two factors are the unique 7/3‑powers in the whole sequence. The authors also show that twelve squares is optimal: any infinite binary word that avoids exponents greater than 7/3 must contain at least twelve squares.

The construction is based on a two‑stage morphic substitution system. The first stage uses the classic Thue‑Morse morphism (0→01, 1→10) to generate a highly non‑repetitive base. The second stage applies carefully designed local replacement rules that target specific short patterns (e.g., 001, 110) and replace them with longer blocks engineered to prevent the creation of new squares. By iterating these rules indefinitely, the resulting limit word stabilises in a state where the only possible squares are the twelve predetermined ones.

A key technical tool is a compression‑expansion analysis. The authors compress the infinite word into blocks of bounded length, compute each block’s minimal period, and verify that the ratio length/period never exceeds 7/3 except for the two explicit 7/3‑powers. If a block’s period were too small, the replacement rules would have been triggered, guaranteeing that no new high‑exponent factor can appear. This method also yields a rigorous impossibility proof: any attempt to reduce the number of squares below twelve inevitably forces the emergence of a factor with exponent >7/3, which the authors demonstrate by exhaustive case analysis of all possible square configurations.

Beyond the binary case, the paper proposes a conjecture for the ternary alphabet: the finite‑repetition threshold should be 7/4, matching the classical repetition threshold for three symbols. Preliminary computational experiments support this conjecture, suggesting that as the alphabet size grows, the FRT converges to the ordinary repetition threshold.

In summary, the authors achieve three major contributions: (1) they define and motivate the finite‑repetition threshold as a natural refinement of the repetition threshold concept; (2) they prove that the binary FRT is 7/3 and exhibit an optimal infinite word with exactly twelve squares and two 7/3‑powers; (3) they provide a methodological framework—morphic constructions combined with compression‑expansion analysis—that can be adapted to larger alphabets, opening a new line of inquiry into the interplay between the quantity of repetitions and the maximal allowed exponent in infinite words.