On Infinite Prefix Normal Words

Prefix normal words are binary words that have no factor with more $1$s than the prefix of the same length. Finite prefix normal words were introduced in [Fici and Lipt'ak, DLT 2011]. In this paper, we study infinite prefix normal words and explore their relationship to some known classes of infinite binary words. In particular, we establish a connection between prefix normal words and Sturmian words, between prefix normal words and abelian complexity, and between prefix normality and lexicographic order.

💡 Research Summary

The paper investigates infinite binary words that satisfy the prefix‑normal property: for every length i, the number of 1’s in the length‑i prefix equals the maximum number of 1’s among all factors of length i. After recalling basic definitions (prefix‑normality, minimum density δ, minimum‑density prefix ι, slope), the authors introduce two constructive operations. The first, flip‑ext, appends the smallest possible block of 0’s followed by a 1 so that the resulting word remains prefix‑normal; repeated application (flip‑ext^ω) yields an infinite word that is densest among all prefix‑normal extensions of the same finite prefix. The second, lazy‑flip‑ext, is parameterised by a target density α; it adds as many 0’s as possible without dropping below α, then a 1. When α equals the original word’s minimum density, the infinite word produced is the sparsest possible extension. Both operations preserve the minimum density and the minimum‑density prefix.

The authors then connect prefix‑normality with several well‑studied families of infinite words. They prove that a Sturmian word w is prefix‑normal if and only if w = 1 c_α, where c_α is the characteristic word of slope α; more generally, any c‑balanced word can be turned prefix‑normal by prefixing a bounded number of 1’s. The Thue‑Morse word is not prefix‑normal, but prefixing two 1’s makes it so; the Champernowne word cannot be made prefix‑normal by any finite number of leading 1’s.

A central contribution is the tight relationship between prefix‑normal forms and abelian complexity. The paper shows that knowing the prefix‑normal form of an infinite word determines its abelian complexity function, and conversely, abelian complexity data can be used to reconstruct the prefix‑normal form. This yields explicit prefix‑normal forms for Sturmian words, Thue‑Morse morphic images, and words generated by uniform morphisms.

In the lexicographic context, the authors compare infinite prefix‑normal words with infinite Lyndon words and with the max/min‑words studied previously. They demonstrate that the four notions (prefix‑normal, Lyndon, max‑word, min‑word) are independent: examples exist where a word satisfies any combination of them.

Finally, the paper establishes criteria for periodicity, ultimate periodicity, and aperiodicity of prefix‑normal words in terms of the minimum density δ and the index ι of the minimum‑density prefix. These criteria allow a simple classification of the infinite words generated by the two operations.

Overall, the work provides a comprehensive theory of infinite prefix‑normal words, linking them to Sturmian sequences, abelian complexity, and lexicographic order, and offering constructive tools (flip‑ext, lazy‑flip‑ext) for generating extremal examples. The results open new avenues for combinatorics on words, formal language theory, and algorithmic applications such as indexing and compression.