Dejean's theorem

Dejean's theorem (formerly Dejean's conjecture) is a statement about repetitions in infinite strings of symbols. It belongs to the field of combinatorics on words; it was conjectured in 1972 by Françoise Dejean and proven in 2009 by Currie and Rampersad and, independently, by Rao.^[1]

Context

In the study of strings, concatenation is seen as analogous to multiplication of numbers. For instance, if ${\displaystyle s}$ is any string, then the concatenation ${\displaystyle ss}$ of two copies of ${\displaystyle s}$ is called the square of ${\displaystyle s}$, and denoted ${\displaystyle s^{2}}$. This exponential notation may also be extended to fractional powers: if ${\displaystyle s}$ has length ${\displaystyle \ell }$, and ${\displaystyle e}$ is a non-negative rational number of the form ${\displaystyle n/\ell }$, then ${\displaystyle s^{e}}$ denotes the string formed by the first ${\displaystyle n}$ characters of the infinite repetition ${\displaystyle sssss\dots }$.^[1]

A square-free word is a string that does not contain any square as a substring. In particular, it avoids repeating the same symbol consecutively, repeating the same pair of symbols, etc. Axel Thue showed that there exists an infinite square-free word using a three-symbol alphabet, the sequence of differences between consecutive elements of the Thue–Morse sequence. However, it is not possible for an infinite two-symbol word (or even a two-symbol word of length greater than three) to be square-free.^[1]

For alphabets of two symbols, however, there do exist infinite cube-free words, words with no substring of the form ${\displaystyle sss}$. One such example is the Thue–Morse sequence itself; another is the Kolakoski sequence. More strongly, the Thue–Morse sequence contains no substring that is a power strictly greater than two.^[1]

In 1972, Dejean investigated the problem of determining, for each possible alphabet size, the threshold between exponents ${\displaystyle e}$ for which there exists an infinite ${\displaystyle e}$-power-free word, and the exponents for which no such word exists. The problem was solved for two-symbol alphabets by the Thue–Morse sequence, and Dejean solved it as well for three-symbol alphabets. She conjectured a precise formula for the threshold exponent for every larger alphabet size;^[2] this formula is Dejean's conjecture, now a theorem.^[1]

Statement

Let ${\displaystyle k}$ be the number of symbols in an alphabet. For every ${\displaystyle k}$, define ${\displaystyle \operatorname {RT} (k)}$, the repeat threshold, to be the infimum of exponents ${\displaystyle e}$ such that there exists an infinite ${\displaystyle e}$-power-free word on a ${\displaystyle k}$-symbol alphabet. Thus, for instance, the Thue–Morse sequence shows that ${\displaystyle \operatorname {RT} (2)=2}$, and an argument based on the Lovász local lemma can be used to show that ${\displaystyle \operatorname {RT} (k)}$ is finite for all ${\displaystyle k}$.^[1]

Then Dejean's conjecture is that the repeat threshold can be calculated by the simple formula^[1][2]

${\displaystyle \operatorname {RT} (k)={\frac {k}{k-1}}}$

except in two exceptional cases:

${\displaystyle \operatorname {RT} (3)={\frac {7}{4}}}$

and

${\displaystyle \operatorname {RT} (4)={\frac {7}{5}}.}$

Progress and proof

Dejean herself proved the conjecture for ${\displaystyle k=3}$.^[2] The case ${\displaystyle k=4}$ was proven by Jean-Jacques Pansiot in 1984.^[3] The next progress was by Moulin Ollagnier in 1992, who proved the conjecture for all alphabet sizes up to ${\displaystyle k\leq 11}$.^[4] This analysis was extended up to ${\displaystyle k\leq 14}$ in 2007 by Mohammad-Noori and Currie.^[5]

In the other direction, also in 2007, Arturo Carpi showed the conjecture to be true for large alphabets, with ${\displaystyle k\geq 33}$.^[6] This reduced the problem to a finite number of remaining cases, which were solved in 2009 and published in 2011 by Currie and Rampersad^[7] and independently by Rao.^[8]

Dejean words

An infinite string that meets Dejean's formula (having no repetitions of exponent above the repetition threshold) is called a Dejean word. Thus, for instance, the Thue–Morse sequence is a Dejean word.

References