A $40$ year old mystery about how patterns are distributed in sequences of three symbols has finally been proven impossible.
Rauzy's Conjecture deals with abelian complexity, or how many different combinations of letters appear in a sequence. This proof shows that you cannot have an infinite sequence of three symbols that has both a constant complexity and independent letter frequencies. It establishes a hard limit on the fundamental randomness of patterns in three-letter alphabets. This result has major implications for symbolic dynamics and the study of quasi-crystals. It marks the end of one of the most famous open problems in the study of mathematical words.
A Proof of Rauzy's Conjecture on Abelian Complexity
arXiv · 2605.01577
A celebrated theorem by Coven and Hedlund (1973) states that Sturmian words are characterized by their abelian complexity: they are precisely the infinite words with rationally independent letter frequencies and constant abelian complexity equal to 2. In this article, we prove a conjecture of Rauzy (1983), showing that there do not exist infinite ternary words with rationally independent letter frequencies and constant abelian complexity equal to 3.