SLIDE 1
Dejean’s conjecture and letter frequency
Jérémie Chalopin Pascal Ochem
LaBRI, Université Bordeaux 1
Mons days 2006
SLIDE 2 Extremal letter frequencies
Σi is the i-letter alphabet {0,1,...,i −1}.
L is a factorial language over Σi.
Frequency of the letter 0 in w : f(w) = |w|0
|w| .
Extremal letter frequencies in L :
|w|→∞min{f(w) | w ∈ L}
|w|→∞max{f(w) | w ∈ L}
SLIDE 3
repetitions with fractional exponent
n n t
t n-repetition
0120 is a 4
3-repetition.
(prefix : 012) 0112011201 is a 5
2-repetition.
(prefix : 0112) A word is q+-free if it contains no repetition of exponent > q.
SLIDE 4 Dejean’s conjecture
A k-word is an infinite
k−1
+
Conjecture (Dejean 1972)
For every k ≥ 5, there exists a k-word.
Proved for 5 ≤ k ≤ 14 and k ≥ 38 [Carpi 2006].
SLIDE 5
Strong forms of Dejean’s conjecture
Conjecture (O. 2005) (1) For every k ≥ 5, there exists a k-word with letter frequency
1 k+1.
(2) For every k ≥ 6, there exists a k-word with letter frequency
1 k−1.
For 5-words, fmax < 103
440 = 0.23409··· < 1 4.
SLIDE 6
Our results
Theorem (Chalopin, O. 2005) (1) There exists a 5-word with letter frequency 1
6.
(2) There exists a 6-word with letter frequency 1
5.
SLIDE 7
Form of k-words with extremal letter frequency
In both cases, 0’s must be regularly spaced : (1) fmin =
1 k+1 :
······0α1 ···αk−1α10β1 ···βk−1β10γ1 ···γk−1γ10······ (2) fmax =
1 k−1 :
······0δ1 ···δk−20σ1 ···σk−20ω1 ···ωk−20······ Where α,β,γ,δ,σ,ω are permutations of [1,2,...,k −1]. NB : For k-words with f = 1
k , 0’s cannot be regularly spaced :
0,1,2,...,k −1,0,2,1,...,0,1,2 = ⇒ it is a repetition of exponent 2k+3
2k
>
k k−1.
SLIDE 8 Minimal letter frequency
5-words with letter frequency 1
6.
Two possible transitions :
- 012341024312 correspond to the transition permutation
2431 (noted 0).
- 012341032143 correspond to the transition permutation
3214 (noted 1). Example : 01001 is coding the word 012341024312034213041234013241023142. = ⇒ It remains to construct an infinite binary code.
SLIDE 9
Minimal letter frequency
One infinite binary code is the fixed point hω(0) of the following morphism h.
h(0) = 010010010100101001001001010100101001001001010010101001001001010 h(1) = 100101001001010100101001001010100101001001001010010010010100101
Useful properties of h : (1) uniform : |h(0)| = |h(1)| = 63. (2) synchronizing : ∀a,b,c ∈ Σ2, ∀s,r ∈ Σ∗
2, if h(ab) = rh(c)s,
then either r = ε and a = c or s = ε and b = c. (3) ∀i ∈ Σ2, h(i) = iwi. (4) ∀i ∈ Σ2, h(i) is the same transition as i.
SLIDE 10 Maximal letter frequency
6-words with letter frequency 1
5.
Two possible transitions :
- 0123405132 correspond to the transition permutation
51324 (noted 0).
- 0123405213 correspond to the transition permutation
52134 (noted 1). One infinite binary code is the fixed point hω(0) of the following 25-uniform morphism m.
m(0) = 0010010100111000110100010 m(1) = 1000100111000100110100011
SLIDE 11 Conclusion
- interest : avoiding repetition & some regularity.
- Other cases are harder :
For 6-words with letter frequency 1
7.
Three possible transitions, and no infinite code using only two of these three transitions.
