Avoiding 3/2-powers over the natural numbers

TL;DR

The paper determines the lexicographically least infinite word over the natural numbers avoiding 3/2-powers, showing it has a compact morphic description and is 6-regular. It also studies the variant avoiding only exact 3/2-powers and shows its structure is closely related.

Key findings

• The lexicographically least 3/2-power-free word \(w_{3/2}\) begins \(01203102130120410214012031021501204102130120310214\cdots\), is 6-regular, and satisfies \(w_{3/2}=\tau(\phi^\omega(3))\) for explicitly defined morphisms \(\phi,\tau\).
• \(w_{3/2}\) is proved 3/2-power-free and lexicographically least: decrementing any nonzero symbol introduces a 3/2-power, with factors such as \(00\), \(020\), \(030\), \(040\), \(101\), and \(202\) appearing in the case analysis.
• The variant \(x_{3/2}\), the lexicographically least word over \(\mathbb N\) avoiding exact 3/2-powers, is also 6-regular and satisfies \(x_{3/2}=\upsilon(\phi^\omega(3))\); it contains squares, but its only square factors are \(00\) and \(11\), so it is overlap-free.
• The proof handles factors \(xyx\) with \(|y|=|x|\) or \(|y|=|x|-1\) via residue cases modulo 10, and modulo 12 for \(x_{3/2}\); one subcase requires examining 30 cases.

Methods

• The authors define an infinite sequence \(v\) via a period-10 pseudoperiodic/self-similar construction, then prove it is 3/2-power-free by showing no factor \(xyx\) with \(|y|=|x|\) or \(|y|=|x|-1\) occurs, using case analysis on \(k \bmod 10\).
• To prove lexicographic minimality, they show that decrementing any nonzero entry of \(v\) (or of \(y\) for \(x_{3/2}\)) creates a 3/2-power, often in the preceding 15 or 30 symbols.
• They give morphism descriptions and use \(k\)-regularity; for the exact-power variant they derive analogous residue cases modulo 12 and use base-6 representations in auxiliary sequences.

Limitations

• The paper states that a deeper understanding of the lexicographically least word avoiding \(\alpha\)-powers for arbitrary \(\alpha\) remains open; computer experiments suggest a similar structure for 4/3-powers, but it is not known whether the lexicographically least word avoiding 5/2-powers uses only finitely many distinct letters.
• Evident from the text: parts of the proof are case-based and “tedious,” with some cases left to the reader (e.g., 30 cases in one verification), which limits transparency.
• In the supplied text, several displayed formulas are marked “formula-not-decoded,” so some details cannot be independently checked from this text alone. (Evident from supplied text, not a stated paper limitation.)