Abelian Repetitions in Sturmian Words
Gabriele Fici
Formal Languages and Automata: Models, Methods and Applications
Naples, Italy – 14 January 2016
G. Fici
Abelian Repetitions in Sturmian Words
Part I
Basics on Sturmian Words
G. Fici
Abelian Repetitions in Sturmian Words
Sturmian words
5
4
3
2
1
2
4
6
8
How to digitally approximate a ray of equation y = αx + ρ?
(we suppose α irrational between 0 and 1)
G. Fici
Abelian Repetitions in Sturmian Words
Sturmian words
5
4
3
2
1
2
4
6
8
How to digitally approximate a ray of equation y = αx + ρ?
(we suppose α irrational between 0 and 1)
All we need is to know if we cross a horizontal line
G. Fici
or not
Abelian Repetitions in Sturmian Words
.
Sturmian words
5
4
3
2
1
2
4
6
8
How to digitally approximate a ray of equation y = αx + ρ?
(we suppose α irrational between 0 and 1)
All we need is to know if we cross a horizontal line
or not
.
We write a if we cross, b if we do not cross, and we get the Sturmian
word sα,ρ of slope α and intercept ρ.
G. Fici
Abelian Repetitions in Sturmian Words
Sturmian words
5
4
3
2
1
2
4
6
8
How to digitally approximate a ray of equation y = αx + ρ?
(we suppose α irrational between 0 and 1)
All we need is to know if we cross a horizontal line
or not
.
We write a if we cross, b if we do not cross, and we get the Sturmian
word sα,ρ of slope α and intercept ρ.
√
In the figure, α = ρ = ( 5 − 1)/2 = ϕ − 1 gives the Fibonacci word:
sϕ−1,ϕ−1 = abaababaabaab · · ·
G. Fici
Abelian Repetitions in Sturmian Words
Sturmian words
So the n-th letter of the Sturmian word sα,ρ is:
b if {ρ + nα} ∈ [0, 1 − α),
an =
a if {ρ + nα} ∈ (1 − α, 1].
Example
n
{ρ + nα}
an
0
1
2
3
4
5
6
7
8
9
.618 .236 .854 .472 .090 .708 .326 .944 .562 .180
a
b
a
a
b
a
b
a
a
Table: The first letters of the Fibonacci word f = sϕ−1,ϕ−1 .
G. Fici
Abelian Repetitions in Sturmian Words
b
Part II
Abelian Powers and Repetitions in
Sturmian Words
G. Fici
Abelian Repetitions in Sturmian Words
Sturmian words
Remarks:
Sturmian words with the same slope have the same factors, so we
will talk of a generic Sturmian word sα of slope (or angle) α;
G. Fici
Abelian Repetitions in Sturmian Words
Sturmian words
Remarks:
Sturmian words with the same slope have the same factors, so we
will talk of a generic Sturmian word sα of slope (or angle) α;
Many combinatorial properties of the factors of sα depend on the
finer arithmetical properties of the irrational α;
G. Fici
Abelian Repetitions in Sturmian Words
Sturmian words
Remarks:
Sturmian words with the same slope have the same factors, so we
will talk of a generic Sturmian word sα of slope (or angle) α;
Many combinatorial properties of the factors of sα depend on the
finer arithmetical properties of the irrational α;
In particular, if α is badly approximable by a rational, then the
maximum exponent (ratio between the length and the period) of a
factor is low;
G. Fici
Abelian Repetitions in Sturmian Words
Sturmian words
Remarks:
Sturmian words with the same slope have the same factors, so we
will talk of a generic Sturmian word sα of slope (or angle) α;
Many combinatorial properties of the factors of sα depend on the
finer arithmetical properties of the irrational α;
In particular, if α is badly approximable by a rational, then the
maximum exponent (ratio between the length and the period) of a
factor is low;
The limit case is α = ϕ − 1 (Fibonacci word), for which the
maximum exponent of a factor is the lowest possible (2 + ϕ);
G. Fici
Abelian Repetitions in Sturmian Words
Sturmian words
Remarks:
Sturmian words with the same slope have the same factors, so we
will talk of a generic Sturmian word sα of slope (or angle) α;
Many combinatorial properties of the factors of sα depend on the
finer arithmetical properties of the irrational α;
In particular, if α is badly approximable by a rational, then the
maximum exponent (ratio between the length and the period) of a
factor is low;
The limit case is α = ϕ − 1 (Fibonacci word), for which the
maximum exponent of a factor is the lowest possible (2 + ϕ);
For a Sturmian word sα , the maximum exponent of a factor is
limited if and only if α has bounded partial quotients and in this
case the exact value comes from a formula involving these partial
quotients.
G. Fici
Abelian Repetitions in Sturmian Words
Sturmian words
Remarks:
Sturmian words with the same slope have the same factors, so we
will talk of a generic Sturmian word sα of slope (or angle) α;
Many combinatorial properties of the factors of sα depend on the
finer arithmetical properties of the irrational α;
In particular, if α is badly approximable by a rational, then the
maximum exponent (ratio between the length and the period) of a
factor is low;
The limit case is α = ϕ − 1 (Fibonacci word), for which the
maximum exponent of a factor is the lowest possible (2 + ϕ);
For a Sturmian word sα , the maximum exponent of a factor is
limited if and only if α has bounded partial quotients and in this
case the exact value comes from a formula involving these partial
quotients.
What can be said for abelian repetitions?
G. Fici
Abelian Repetitions in Sturmian Words
Abelian powers
The Parikh vector (a.k.a composition vector) of a word w over the
ordered alphabet Σ = {a1 , a2 , . . . , aσ } is the vector
P (w) = (|w|a1 , |w|a2 , . . . , |w|aσ )
counting the occurrences of each letter of Σ in w.
For example, the Parikh vector of w = caa over Σ = {a, b, c} is
P (w) = (2, 0, 1).
G. Fici
Abelian Repetitions in Sturmian Words
Abelian powers
The Parikh vector (a.k.a composition vector) of a word w over the
ordered alphabet Σ = {a1 , a2 , . . . , aσ } is the vector
P (w) = (|w|a1 , |w|a2 , . . . , |w|aσ )
counting the occurrences of each letter of Σ in w.
For example, the Parikh vector of w = caa over Σ = {a, b, c} is
P (w) = (2, 0, 1).
An abelian n-power is a word of the form v1 v2 · · · vn where all the vi ’s
have the same Parikh vector.
G. Fici
Abelian Repetitions in Sturmian Words
Abelian powers
The Parikh vector (a.k.a composition vector) of a word w over the
ordered alphabet Σ = {a1 , a2 , . . . , aσ } is the vector
P (w) = (|w|a1 , |w|a2 , . . . , |w|aσ )
counting the occurrences of each letter of Σ in w.
For example, the Parikh vector of w = caa over Σ = {a, b, c} is
P (w) = (2, 0, 1).
An abelian n-power is a word of the form v1 v2 · · · vn where all the vi ’s
have the same Parikh vector.
An abelian 2-power is called an abelian square.
G. Fici
Abelian Repetitions in Sturmian Words
Abelian powers
The Parikh vector (a.k.a composition vector) of a word w over the
ordered alphabet Σ = {a1 , a2 , . . . , aσ } is the vector
P (w) = (|w|a1 , |w|a2 , . . . , |w|aσ )
counting the occurrences of each letter of Σ in w.
For example, the Parikh vector of w = caa over Σ = {a, b, c} is
P (w) = (2, 0, 1).
An abelian n-power is a word of the form v1 v2 · · · vn where all the vi ’s
have the same Parikh vector.
An abelian 2-power is called an abelian square.
Examples of abelian squares: papa, esse, termometro.
G. Fici
Abelian Repetitions in Sturmian Words
Abelian period
Definition (S. Constantinescu, L. Ilie, 2006)
An integer p is an abelian period for a word w over Σ if w can be written
as
w = u0 u1 · · · un un+1 ,
where n ≥ 1 and for every 1 ≤ i ≤ n all the ui ’s have the same Parikh
vector P, with sum of components equal to p, and the Parikh vectors P 0
of u0 and P 00 of un+1 are properly contained in P.
u0
u1
u2
P0
P
P
···
un
P
un+1
P 00
Example
2 is the smallest abelian period of w = abaab = a · ba · ab · ε.
G. Fici
Abelian Repetitions in Sturmian Words
Abelian repetition
Definition
We say that w is an abelian repetition of abelian period m and abelian
exponent k = |w|/m if one can write
w = u0 u1 · · · un un+1
where n ≥ 2 and for every 1 ≤ i ≤ n all the ui ’s have the same Parikh
vector P, with sum of components equal to m, and the Parikh vectors P 0
of u0 and P 00 of un+1 are properly contained in P.
G. Fici
Abelian Repetitions in Sturmian Words
Abelian repetition
Definition
We say that w is an abelian repetition of abelian period m and abelian
exponent k = |w|/m if one can write
w = u0 u1 · · · un un+1
where n ≥ 2 and for every 1 ≤ i ≤ n all the ui ’s have the same Parikh
vector P, with sum of components equal to m, and the Parikh vectors P 0
of u0 and P 00 of un+1 are properly contained in P.
An abelian power is therefore an abelian repetition with u0 = un+1 = ε.
A word with abelian exponent k = 1, i.e., with minimum abelian period
equal to |w|, is called a degenerated abelian power.
G. Fici
Abelian Repetitions in Sturmian Words
Abelian repetition
Definition
We say that w is an abelian repetition of abelian period m and abelian
exponent k = |w|/m if one can write
w = u0 u1 · · · un un+1
where n ≥ 2 and for every 1 ≤ i ≤ n all the ui ’s have the same Parikh
vector P, with sum of components equal to m, and the Parikh vectors P 0
of u0 and P 00 of un+1 are properly contained in P.
An abelian power is therefore an abelian repetition with u0 = un+1 = ε.
A word with abelian exponent k = 1, i.e., with minimum abelian period
equal to |w|, is called a degenerated abelian power.
Example
w = abaab is an abelian repetition of period 2 and exponent 5/2.
G. Fici
Abelian Repetitions in Sturmian Words
Abelian Powers in Sturmian Words
G. Richomme, K. Saari and L. Q. Zamboni (J. Lond. Math. Soc., 2011)
proved that in any Sturmian word start abelian powers of arbitrary
exponent at every position.
G. Fici
Abelian Repetitions in Sturmian Words
Abelian Powers in Sturmian Words
G. Richomme, K. Saari and L. Q. Zamboni (J. Lond. Math. Soc., 2011)
proved that in any Sturmian word start abelian powers of arbitrary
exponent at every position.
We show what are the maximum exponents of abelian powers and
abelian repetitions of period m for every given m.
G. Fici
Abelian Repetitions in Sturmian Words
Abelian Powers in Sturmian Words
Theorem
Let sα be a Sturmian word with slope α, and m a positive integer. Then
sα contains an abelian power of period m and exponent k ≥ 2 if and only
if kmαk < k1 , where kmαk = min({mα}, {−mα}) is the distance of mα
form the closest integer.
Corollary
Let sα be a Sturmian word with slope α, and m a positive integer. Then
the maximum exponent km of an abelian power of period m in sα is the
largest k such that kmαk < 1/k, i.e.,
1
km =
.
kmαk
G. Fici
Abelian Repetitions in Sturmian Words
Abelian Powers in Sturmian Words
Theorem
Let sα be a Sturmian word with slope α, and m a positive integer. Then
sα contains an abelian power of period m and exponent k ≥ 2 if and only
if kmαk < k1 , where kmαk = min({mα}, {−mα}) is the distance of mα
form the closest integer.
Corollary
Let sα be a Sturmian word with slope α, and m a positive integer. Then
the maximum exponent km of an abelian power of period m in sα is the
largest k such that kmαk < 1/k, i.e.,
1
km =
.
kmαk
m
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21
km
2 4 6 2 11 3 3 17 2 5
4
2 29 2
3
8
2
8
3
2 46
Table: The first values of the maximum exponent km of an abelian power of
period m in the Fibonacci word f = sϕ−1,ϕ−1 .
G. Fici
Abelian Repetitions in Sturmian Words
From Number Theory to Abelian Powers
We recall a classical result:
Theorem (A. Hurwitz, 1891)
Any irrational α has an infinity of rational approximations n/m which
satisfy
n
1
− α < √ 2 .
m
5m
√
Moreover, the √
constant 5 is best possible. Indeed, if α = ϕ − 1, then
for every A > 5 the inequality
n
1
− α <
m
Am2
has only a finite number of solutions n/m.
G. Fici
Abelian Repetitions in Sturmian Words
From Number Theory to Abelian Powers
For a general irrational α, the infimum λ = λ(α) such that for every
A > λ, the inequality
n
1
− α <
m
Am2
has only a finite number of solutions, is called the Lagrange constant of
α. (It is finite if and only if α has bounded partial quotients.)
G. Fici
Abelian Repetitions in Sturmian Words
From Number Theory to Abelian Powers
For a general irrational α, the infimum λ = λ(α) such that for every
A > λ, the inequality
n
1
− α <
m
Am2
has only a finite number of solutions, is called the Lagrange constant of
α. (It is finite if and only if α has bounded partial quotients.)
The set of finite Lagrange constants of the irrationals is called the
Lagrange spectrum.
G. Fici
Abelian Repetitions in Sturmian Words
From Number Theory to Abelian Powers
For a general irrational α, the infimum λ = λ(α) such that for every
A > λ, the inequality
n
1
− α <
m
Am2
has only a finite number of solutions, is called the Lagrange constant of
α. (It is finite if and only if α has bounded partial quotients.)
The set of finite Lagrange constants of the irrationals is called the
Lagrange spectrum.
√ √
Its
√ structure is still not well understood. Its least elements are 5, 8,
221/5 (so the beginning is denumerable) but it also contains all the
real numbers larger than the Freiman’s constant
√
2221564096 + 283748 462
= 4.5278295661 . . .
491993569
G. Fici
Abelian Repetitions in Sturmian Words
From Number Theory to Abelian Powers
Since in any Sturmian word, the abelian exponent of factors can be
arbitrarily large, we give the following:
Definition
The abelian critical exponent of a Sturmian word sα of slope α is defined
as
km
k0
= lim sup m .
A(sα ) = lim sup
m→∞ m
m→∞ m
0
where km (resp. km
) is the maximum exponent of an abelian power
(resp. abelian repetition) of period m in sα .
G. Fici
Abelian Repetitions in Sturmian Words
From Number Theory to Abelian Powers
Theorem
The abelian critical exponent A(sα ) of a Sturmian word sα equals the
Lagrange constant λ(α) of its slope.
In particular, the following are equivalent:
A(sα ) is finite;
sα is β-power-free for some β ≥ 2;
α has bounded partial quotients.
Furthermore, one has:
A(sα ) = lim sup ([ai+1 ; ai+2 , . . .] + [0; ai , ai−1 , . . . , a1 ]) .
i→∞
For example, the Fibonacci
word has the smallest possible abelian critical
√
exponent, that is 5.
G. Fici
Abelian Repetitions in Sturmian Words
Abelian Repetitions in the Fibonacci Word
J. Currie and K. Saari (RAIRO ITA, 2009) proved the following:
Theorem
The smallest period of any factor of the Fibonacci word is a Fibonacci
number.
G. Fici
Abelian Repetitions in Sturmian Words
Abelian Repetitions in the Fibonacci Word
J. Currie and K. Saari (RAIRO ITA, 2009) proved the following:
Theorem
The smallest period of any factor of the Fibonacci word is a Fibonacci
number.
We prove that this result can be generalized to the abelian setting:
Theorem
The smallest abelian period of any factor of the Fibonacci word is a
Fibonacci number.
The proof is not trivial and involves several of the previous results.
G. Fici
Abelian Repetitions in Sturmian Words
Abelian Repetitions in the Fibonacci Word
Let f0 = b, f1 = a, and fj = fj−1 fj−2 for every j > 1 be the sequence
of Fibonacci finite words (for every j > 1, fj is the prefix of the
Fibonacci word of length Fj ).
G. Fici
Abelian Repetitions in Sturmian Words
Abelian Repetitions in the Fibonacci Word
Let f0 = b, f1 = a, and fj = fj−1 fj−2 for every j > 1 be the sequence
of Fibonacci finite words (for every j > 1, fj is the prefix of the
Fibonacci word of length Fj ).
Corollary
For j ≥ 3, the (smallest) abelian period of the word fj is the n-th
Fibonacci number Fn , where
(
bj/2c
if j = 0, 1, 2 mod 4;
n=
bj/2c + 1 if j = 3 mod 4.
For example, the abelian period of the word f4 = abaab is
2 = F2 = Fb4/2c , since one can write f4 = a · ba · ab; the abelian period
of f7 = abaababaabaababaababa is 5 = F4 = F1+b7/2c .
G. Fici
Abelian Repetitions in Sturmian Words
Abelian Repetitions in the Fibonacci Word
Let f0 = b, f1 = a, and fj = fj−1 fj−2 for every j > 1 be the sequence
of Fibonacci finite words (for every j > 1, fj is the prefix of the
Fibonacci word of length Fj ).
Corollary
For j ≥ 3, the (smallest) abelian period of the word fj is the n-th
Fibonacci number Fn , where
(
bj/2c
if j = 0, 1, 2 mod 4;
n=
bj/2c + 1 if j = 3 mod 4.
For example, the abelian period of the word f4 = abaab is
2 = F2 = Fb4/2c , since one can write f4 = a · ba · ab; the abelian period
of f7 = abaababaabaababaababa is 5 = F4 = F1+b7/2c .
j
3
4
5
6
7
8
9
10
11
12
13
14
15
16
a. p. of fj
2
2
2
3
5
5
5
8
13 13 13 21 34 34
F2 F2 F2 F3 F4 F4 F4 F5 F6 F6 F6 F7 F8 F8
G. Fici
Abelian Repetitions in Sturmian Words
Open Problems
We investigated combinatorial properties of abelian powers and abelian
repetitions in Sturmian words and in particular in the Fibonacci word.
We gave explicit tight bounds on the lengths and the exponents of
abelian powers depending on the slope of the Sturmian word.
A natural direction of investigation is whether it is possible to extend
these results to larger classes of words, e.g. binary rotation words or
Arnoux-Rauzy words.
G. Fici
Abelian Repetitions in Sturmian Words
Part III
Abelian-square Rich Words
G. Fici
Abelian Repetitions in Sturmian Words
Abelian Square Factors
Remark
A word of length n can contain Θ(n2 ) distinct factors. Therefore, a word
of length n contains O(n2 ) distinct abelian square factors.
G. Fici
Abelian Repetitions in Sturmian Words
Abelian Square Factors
Remark
A word of length n can contain Θ(n2 ) distinct factors. Therefore, a word
of length n contains O(n2 ) distinct abelian square factors.
Question: Can a word of length n contain Θ(n2 ) distinct abelian square
factors?
G. Fici
Abelian Repetitions in Sturmian Words
Abelian Square Factors
Remark
A word of length n can contain Θ(n2 ) distinct factors. Therefore, a word
of length n contains O(n2 ) distinct abelian square factors.
Question: Can a word of length n contain Θ(n2 ) distinct abelian square
factors?
Answer: YES.
Example
The word wn = an ban ban contains Θ(n2 ) distinct abelian square factors.
Proof: For any 0 ≤ i, j ≤ n, if the factor ai ban baj has even length, then
it is an abelian square.
Since the number of possible choices for the pair (i, j) is quadratic in n,
we are done.
G. Fici
Abelian Repetitions in Sturmian Words
Abelian-square Rich Words
Definition
An infinite word w is abelian-square rich if and only if there exists a
positive constant C such that for every n sufficiently large one has
1
|Fact(w) ∩ Σn |
X
{# abelian square factors of v} ≥ Cn2 .
v∈Fact(w)∩Σn
G. Fici
Abelian Repetitions in Sturmian Words
Abelian-square Rich Words
Definition
An infinite word w is abelian-square rich if and only if there exists a
positive constant C such that for every n sufficiently large one has
1
|Fact(w) ∩ Σn |
X
{# abelian square factors of v} ≥ Cn2 .
v∈Fact(w)∩Σn
In other words, a word is abelian square rich if, on average, any factor
contains a number of abelian square factors that is quadratic in its
length.
G. Fici
Abelian Repetitions in Sturmian Words
Abelian-square Rich Words
Definition
An infinite word w is abelian-square rich if and only if there exists a
positive constant C such that for every n sufficiently large one has
1
|Fact(w) ∩ Σn |
X
{# abelian square factors of v} ≥ Cn2 .
v∈Fact(w)∩Σn
In other words, a word is abelian square rich if, on average, any factor
contains a number of abelian square factors that is quadratic in its
length.
Christodoulakis et al. (TCS,
√ 2014) proved that a binary word of length n
contains, on average, Θ(n n) distinct abelian square factors, hence an
infinite binary random word is almost surely not abelian-square rich.
G. Fici
Abelian Repetitions in Sturmian Words
Linearly Recurrent Words
Recall that the recurrence index Rw (n) of an infinite word w is the least
integer m (if any exists) such that every factor of w of length m contains
all factors of w of length n.
G. Fici
Abelian Repetitions in Sturmian Words
Linearly Recurrent Words
Recall that the recurrence index Rw (n) of an infinite word w is the least
integer m (if any exists) such that every factor of w of length m contains
all factors of w of length n.
If the recurrence index is defined for every n, the infinite word w is called
uniformly recurrent and the function Rw (n) the recurrence function of w.
G. Fici
Abelian Repetitions in Sturmian Words
Linearly Recurrent Words
Recall that the recurrence index Rw (n) of an infinite word w is the least
integer m (if any exists) such that every factor of w of length m contains
all factors of w of length n.
If the recurrence index is defined for every n, the infinite word w is called
uniformly recurrent and the function Rw (n) the recurrence function of w.
A uniformly recurrent word w is called linearly recurrent if the ratio
Rw (n)/n is bounded.
G. Fici
Abelian Repetitions in Sturmian Words
Linearly Recurrent Words
Recall that the recurrence index Rw (n) of an infinite word w is the least
integer m (if any exists) such that every factor of w of length m contains
all factors of w of length n.
If the recurrence index is defined for every n, the infinite word w is called
uniformly recurrent and the function Rw (n) the recurrence function of w.
A uniformly recurrent word w is called linearly recurrent if the ratio
Rw (n)/n is bounded.
Given a linearly recurrent word w, the real number
rw = lim supn→∞ Rw (n)/n is called the recurrence quotient of w.
G. Fici
Abelian Repetitions in Sturmian Words
Linearly Recurrent Words
Recall that the recurrence index Rw (n) of an infinite word w is the least
integer m (if any exists) such that every factor of w of length m contains
all factors of w of length n.
If the recurrence index is defined for every n, the infinite word w is called
uniformly recurrent and the function Rw (n) the recurrence function of w.
A uniformly recurrent word w is called linearly recurrent if the ratio
Rw (n)/n is bounded.
Given a linearly recurrent word w, the real number
rw = lim supn→∞ Rw (n)/n is called the recurrence quotient of w.
Example
Every Sturmian word is uniformly recurrent. A Sturmian word is linearly
recurrent if and only if its slope has bounded partial quotients.
G. Fici
Abelian Repetitions in Sturmian Words
Abelian-square Rich Words
Given a finite or infinite word w, we let ASFw (n) denote the number of
abelian square factors of w of length n. Of course, ASFw (n) = 0 if n is
odd, so this quantity is significant only for even values of n.
G. Fici
Abelian Repetitions in Sturmian Words
Abelian-square Rich Words
Given a finite or infinite word w, we let ASFw (n) denote the number of
abelian square factors of w of length n. Of course, ASFw (n) = 0 if n is
odd, so this quantity is significant only for even values of n.
The following lemma is a consequence of the definition of linearly
recurrent word.
Lemma
Let w be a linearly recurrent word. IfP
there exists a constant C such that
for every n sufficiently large one has m≤n ASFw (m) ≥ Cn2 , then w is
abelian-square rich.
G. Fici
Abelian Repetitions in Sturmian Words
Uniformly Abelian-square Rich Words
Definition
An infinite word w is uniformly abelian-square rich if and only if there
exists a positive constant C such that for every n sufficiently large one
has
inf
{# abelian square factors of v} ≥ Cn2 .
n
v∈Fact(w)∩Σ
G. Fici
Abelian Repetitions in Sturmian Words
Uniformly Abelian-square Rich Words
Definition
An infinite word w is uniformly abelian-square rich if and only if there
exists a positive constant C such that for every n sufficiently large one
has
inf
{# abelian square factors of v} ≥ Cn2 .
n
v∈Fact(w)∩Σ
In other words, in a uniformly abelian-square rich word every factor
contains a quadratic number of abelian squares.
G. Fici
Abelian Repetitions in Sturmian Words
Uniformly Abelian-square Rich Words
Definition
An infinite word w is uniformly abelian-square rich if and only if there
exists a positive constant C such that for every n sufficiently large one
has
inf
{# abelian square factors of v} ≥ Cn2 .
n
v∈Fact(w)∩Σ
In other words, in a uniformly abelian-square rich word every factor
contains a quadratic number of abelian squares.
Clearly, if a word is uniformly abelian-square rich, then it is also
abelian-square rich, but the converse is not always true.
G. Fici
Abelian Repetitions in Sturmian Words
Uniformly Abelian-square Rich Words
Definition
An infinite word w is uniformly abelian-square rich if and only if there
exists a positive constant C such that for every n sufficiently large one
has
inf
{# abelian square factors of v} ≥ Cn2 .
n
v∈Fact(w)∩Σ
In other words, in a uniformly abelian-square rich word every factor
contains a quadratic number of abelian squares.
Clearly, if a word is uniformly abelian-square rich, then it is also
abelian-square rich, but the converse is not always true.
However, in the case of linearly recurrent words, the two definitions are
equivalent:
Lemma
If w is abelian-square rich and linearly recurrent, then it is uniformly
abelian-square rich.
G. Fici
Abelian Repetitions in Sturmian Words
The Thue-Morse Words
Let
t = 011010011001011010010110 · · ·
be the Thue-Morse word, i.e., the fixed point of the uniform substitution
µ : 0 7→ 01, 1 7→ 10.
The Thue-Morse word t is linearly recurrent.
G. Fici
Abelian Repetitions in Sturmian Words
The Thue-Morse Words
Let
t = 011010011001011010010110 · · ·
be the Thue-Morse word, i.e., the fixed point of the uniform substitution
µ : 0 7→ 01, 1 7→ 10.
The Thue-Morse word t is linearly recurrent.
Proposition
The Thue-Morse word t is abelian-square rich; hence it is uniformly
abelian-square rich.
G. Fici
Abelian Repetitions in Sturmian Words
The Thue-Morse Words
Let
t = 011010011001011010010110 · · ·
be the Thue-Morse word, i.e., the fixed point of the uniform substitution
µ : 0 7→ 01, 1 7→ 10.
The Thue-Morse word t is linearly recurrent.
Proposition
The Thue-Morse word t is abelian-square rich; hence it is uniformly
abelian-square rich.
The proof uses Brlek’s recursive formula for the factor complexity of t
and the following result:
Lemma (Cassaigne, Fici, Sciortino, Zamboni, 2015)
For every n ≥ 2, at least 1/3 of the factors of t of length n start and end
with the same letter, resp. with different letters.
G. Fici
Abelian Repetitions in Sturmian Words
Sturmian Words
Recall that a (finite or infinite) word w over Σ = {a, b} is balanced if and
only if for any u, v factors of w of the same length, one has
||u|a − |v|a | ≤ 1.
A Sturmian word is in fact an infinite balanced aperiodic word over
Σ = {a, b}.
G. Fici
Abelian Repetitions in Sturmian Words
Sturmian Words
Recall that a (finite or infinite) word w over Σ = {a, b} is balanced if and
only if for any u, v factors of w of the same length, one has
||u|a − |v|a | ≤ 1.
A Sturmian word is in fact an infinite balanced aperiodic word over
Σ = {a, b}.
Lemma
Let w be a finite balanced word over Σ. Then for every n > 0,
P (w) = (0, 0) mod n if and only if w is an abelian n-power.
G. Fici
Abelian Repetitions in Sturmian Words
Sturmian Words
Proposition
Let sα be any Sturmian word of angle α. For every positive even n, let
In = {{−iα} | 1 ≤ i ≤ n}. Then
(
#{x ∈ In | x ≤ {−nα}} if bnαc is even;
ASFsα (n) =
#{x ∈ In | x ≥ {−nα}} if bnαc is odd.
Example
We have I6 = {0.382, 0.764, 0.146, 0.528, 0.910, 0.292} (values are
approximated) and 6α ' 3.708, so b6αc is odd. Thus, there are 5
elements in I6 that are ≥ {−6α}, so by the formula there are 5 abelian
square factors of length 6.
G. Fici
Abelian Repetitions in Sturmian Words
Sturmian Words
Here are the first values of the sequence ASFF (n) for the Fibonacci word.
n
ASFF (n)
0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30
0 1 3 5 1
9
5
5
15
3
13 13
5
25
9
15
Table: The first values of the sequence ASFF (n) of the number of abelian
square factors of length n in the Fibonacci word f = sϕ−1,ϕ−1 .
G. Fici
Abelian Repetitions in Sturmian Words
Uniformly Distributed Sequences
Let ω = (xn )n≥0 be a given sequence of real numbers. For a positive
integer N and E ⊂ [0, 1), we define A(E; N ; ω) as the number of terms
xn , 0 ≤ n ≤ N , for which {xn } ∈ E.
If there is no risk of confusion, we will write A(E; N ) instead of
A(E; N ; ω).
G. Fici
Abelian Repetitions in Sturmian Words
Uniformly Distributed Sequences
Let ω = (xn )n≥0 be a given sequence of real numbers. For a positive
integer N and E ⊂ [0, 1), we define A(E; N ; ω) as the number of terms
xn , 0 ≤ n ≤ N , for which {xn } ∈ E.
If there is no risk of confusion, we will write A(E; N ) instead of
A(E; N ; ω).
Definition
The sequence ω = (xn )n≥0 of real numbers is said to be uniformly
distributed modulo 1 if and only if for every pair a, b of real numbers with
0 ≤ a < b ≤ 1 we have
lim
N →∞
A([a, b); N ; ω)
= b − a.
N
G. Fici
Abelian Repetitions in Sturmian Words
Uniformly Distributed Sequences
Definition
Let x0 , x1 , . . . , xN be a finite sequence of real numbers. The number
A([γ, δ); N )
DN = DN (x0 , x1 , . . . , xN ) = sup − (δ − γ)
N
0≤γ<δ≤1
is called the discrepancy of the given sequence. For an infinite sequence
ω of real numbers the discrepancy DN (ω) is the discrepancy of the initial
segment formed by the first N + 1 terms of ω.
Theorem
The sequence ω is uniformly distributed modulo 1 if and only if
limN →∞ DN (ω) = 0.
G. Fici
Abelian Repetitions in Sturmian Words
Uniformly Distributed Sequences
Theorem
Suppose the irrational α = [a0 ; a1 , . . .] has partial quotients bounded by
K. Then the discrepancy DN (ω) of ω = (nα) satisfies
N DN (ω) = O(log N ). More exactly, we have
1
K
N DN (ω) ≤ 3 +
+
log N.
(1)
log ϕ log(K + 1)
G. Fici
Abelian Repetitions in Sturmian Words
Abelian-square Rich Sturmian Words
Using the previous result, we proved the following
Theorem
Let sα be a Sturmian word of slope α such that α has bounded partial
quotients. Then there exists
P a positive constant C such that for every n
sufficiently large one has m≤n ASFsα (m) ≥ Cn2 .
Corollary
Let sα be a Sturmian word of slope α. If sα is β-power free for some
β > 2, then sα is uniformly abelian-square rich.
G. Fici
Abelian Repetitions in Sturmian Words
Conclusions
We proved that the Thue-Morse is uniformly abelian-square rich. We
think that the technique we used for the proof can be generalized to
some extent, and could be used, for example, to prove that a class of
fixed points of uniform substitutions are uniformly abelian-square rich.
G. Fici
Abelian Repetitions in Sturmian Words
Conclusions
We also proved that Sturmian words that are β-power free are uniformly
abelian-square rich. The proof we gave is based on a classical result on
the discrepancy of the uniformly distributed modulo 1 sequence (nα)n≥0 ,
where α is the angle of the Sturmian word. To the best of our
knowledge, this is the first application of this result to the theory of
Sturmian words, and we think that the correspondence we have shown
might be useful for deriving other results on Sturmian words.
The natural question that then arises is whether the hypothesis of
power-freeness is necessary for a Sturmian word being (uniformly)
abelian-square rich. We leave open the question to determine whether sα
is not uniformly abelian-square rich nor abelian-square rich in the case
when α has unbounded partial quotients.
G. Fici
Abelian Repetitions in Sturmian Words
Conclusions
We mostly investigated binary words in this paper. We conjecture that
binary words have the largest number of abelian square factors. More
precisely, we propose the following conjecture.
Conjecture
If a word of length n contains k distinct abelian square factors, then
there exists a binary word of length n containing at least k distinct
abelian square factors.
G. Fici
Abelian Repetitions in Sturmian Words
Conclusions
Two abelian squares are inequivalent if they have different Parikh vectors.
Sturmian words only have a linear number of inequivalent
abelian squares.
√
Nevertheless, a word of length n can contain Θ(n n) inequivalent
abelian squares. Computations support the following conjecture:
Conjecture
√
A word of length n contains O(n n) inequivalent abelian squares.
G. Fici
Abelian Repetitions in Sturmian Words
THANK YOU
G. Fici
Abelian Repetitions in Sturmian Words