combinatorics on words & some applications in number theory

Combinatorics on Words

&

Some Applications in Number Theory

Amy Glen

School of Chemical & Mathematical Sciences

Murdoch University, Perth, Australia

[email protected]://wwwstaff.murdoch.edu.au/∼aglen

Groups & Combinatorics Seminar series @ UWA

Amy Glen (Murdoch University) CoW & Number Theory November 17, 2009 1 / 51

Outline

1 Combinatorics on WordsSturmian & Episturmian Words

2 Some Applications in Number TheoryContinued Fractions & Sturmian WordsDistribution modulo 1 & Sturmian WordsTranscendental Numbers



Outline





Starting point: Combinatorics on words


Number Theory

algebra

Free Groups, SemigroupsMatrices

RepresentationsBurnside Problems

Discrete

Dynamical Systems

TopologyTheoretical Physics

Theoretical

Computer Science

AlgorithmicsAutomata TheoryComputability

Codes

Logic

Probability Theory

Biology

DNA sequencing, Patterns

Discrete Geometry





Number Theory

algebra



Discrete

Dynamical Systems


Theoretical

Computer Science


Codes

Logic

Probability Theory

Biology


Discrete Geometry

A word w is a finite or infinite sequence of symbols (letters) taken from anon-empty finite set A (alphabet).





Number Theory

algebra



Discrete

Dynamical Systems


Theoretical

Computer Science


Codes

Logic

Probability Theory

Biology


Discrete Geometry

A word w is a finite or infinite sequence of symbols (letters) taken from anon-empty finite set A (alphabet).

Example with A = {a, b, c}: w = abca, w∞ = abcaabcaabca · · ·



Combinatorics on words: A brief history

Relatively new area of Discrete Mathematics





Early 1900’s: First investigations by Axel Thue (repetitions in words)






1938: Marston Morse & Gustav Hedlund

Initiated the formal development of symbolic dynamics.








This work marked the beginning of the study of words.








This work marked the beginning of the study of words.

1960’s: Systematic study initiated by M.P. Schützenberger.



Combinatorics on words: Complexity

Most commonly studied words are those which satisfy one or morestrong regularity properties; for instance, words containing manyrepetitions or palindromes.





A palindrome is a word that reads the same backwards as forwards.

Examples: eye, civic, radar, glenelg (Adelaide suburb).







The extent to which a word exhibits strong regularity properties isgenerally inversely proportional to its “complexity”.








Basic measure: Number of distinct blocks (factors) of each lengthoccurring in the word.








Basic measure: Number of distinct blocks (factors) of each lengthoccurring in the word.

Example: w = abca has 9 distinct factors:

a, b, c ,︸︷︷︸

1

ab, bc , ca,︸︷︷︸

2

abc , bca,︸︷︷︸

3

abca.︸︷︷︸

4


Combinatorics on Words Sturmian & Episturmian Words

Sturmian words

Theorem (Morse-Hedlund 1940)

An infinite word w is ultimately periodic if and only if w has less than n + 1distinct factors of length n for some n.



Sturmian words



Sturmian words

Aperiodic infinite words of minimal complexity – exactly n + 1 distinctfactors of length n for each n.



Sturmian words



Sturmian words


Pioneering work by Morse & Hedlund in 1940 (symbolic dynamics).



Sturmian words



Sturmian words



Low complexity accounts for many interesting features, as it inducescertain regularities, without periodicity.



Sturmian words



Sturmian words




Points of view: combinatorial; algebraic; geometric.



Sturmian words



Sturmian words





References in: Combinatorics, Symbolic Dynamics, Number Theory,Discrete Geometry, Theoretical Physics, Theoretical Computer Science.



Sturmian words



Sturmian words





References in: Combinatorics, Symbolic Dynamics, Number Theory,Discrete Geometry, Theoretical Physics, Theoretical Computer Science.

Numerous equivalent definitions & characterisations . . .



Christoffel words: A special family of finite Sturmian words

Lower Christoffel word of slope 35



Christoffel words: Construction






a

L(5,3) = a





a a

L(5,3) = aa





a a

b

L(5,3) = aab





a a

ba

L(5,3) = aaba





a a

ba a

L(5,3) = aabaa





a a

ba a

b

L(5,3) = aabaab





a a

ba a

ba

L(5,3) = aabaaba





a a

ba a

ba

b

L(5,3) = aabaabab




Lower & Upper Christoffel words of slope 35

a a

a a

a

b

b

b

a

a a

a a

b

b

b

L(5,3) = aabaabab U(5,3) = babaabaa



From Christoffel words to Sturmian words

Sturmian words: Obtained *similarly* by replacing the line segment by ahalf-line:

y = αx + ρ with irrational α ∈ (0, 1), ρ ∈ R.



From Christoffel words to Sturmian words

Sturmian words: Obtained *similarly* by replacing the line segment by ahalf-line:

y = αx + ρ with irrational α ∈ (0, 1), ρ ∈ R.

Example: y =√

5−12 x −→ Fibonacci word

a a

a a

a

a a

b

b

b

b

f = abaababaabaababaaba · · · (note: disregard 1st a in construction)

Standard Sturmian word of slope√

5−12 , golden ratio conjugate



Christoffel words: Historical notes

Before the 20th century:

J. Bernoulli, 1771 (Astronomy)

A. Markoff, 1882 (continued fractions)

E. Christoffel, 1871, 1888 (Cayley graphs)



Christoffel words: Historical notes

Before the 20th century:

J. Bernoulli, 1771 (Astronomy)

A. Markoff, 1882 (continued fractions)

E. Christoffel, 1871, 1888 (Cayley graphs)

After the 20th century:

J. Berstel, 1990

J.-P. Borel & F. Laubie, 1993



Christoffel words: Properties

Examples

Slope q/p 3/4 4/3 7/4 5/7

L(p, q) aababab abababb aabaabaabab aababaababab

U(p, q) bababaa bbababa babaabaabaa bababaababaa




Examples

Slope q/p 3/4 4/3 7/4 5/7



Properties

L(p, q) = awb ⇐⇒ U(p, q) = bwa




Examples

Slope q/p 3/4 4/3 7/4 5/7



Properties


|L(p, q)|a = p, |L(p, q)|b = q =⇒ |L(p, q)| = p + q




Examples

Slope q/p 3/4 4/3 7/4 5/7



Properties



L(p, q) is the reversal of U(p, q)




Examples

Slope q/p 3/4 4/3 7/4 5/7



Properties



L(p, q) is the reversal of U(p, q)

Christoffel words are of the form awb, bwa where w is a palindrome.



Christoffel words & palindromes

Theorem (folklore)

A finite word w is a Christoffel word if and only if w = apb or w = bpawhere p = Pal(v) for some word v over {a, b}.




Theorem (folklore)


Pal is the iterated palindromic closure function:

Pal(ε) = ε (empty word) and Pal(wx) = (Pal(w)x)+

for any word w and letter x .




Theorem (folklore)


Pal is the iterated palindromic closure function:

Pal(ε) = ε (empty word) and Pal(wx) = (Pal(w)x)+

for any word w and letter x .

v+: Unique shortest palindrome beginning with v .



Palindromic closure: Examples

(race)+ =




(race)+ = race




(race)+ = race car




(race)+ = race car

(tie)+ =




(race)+ = race car

(tie)+ = tie




(race)+ = race car

(tie)+ = tie it




(race)+ = race car

(tie)+ = tie it

(tops)+ =




(race)+ = race car

(tie)+ = tie it

(tops)+ = top s




(race)+ = race car

(tie)+ = tie it

(tops)+ = top spot




(race)+ = race car

(tie)+ = tie it

(tops)+ = top spot

Pal(aba) =




(race)+ = race car

(tie)+ = tie it

(tops)+ = top spot

Pal(aba) = a




(race)+ = race car

(tie)+ = tie it

(tops)+ = top spot

Pal(aba) = a b




(race)+ = race car

(tie)+ = tie it

(tops)+ = top spot

Pal(aba) = a b a




(race)+ = race car

(tie)+ = tie it

(tops)+ = top spot

Pal(aba) = a b a a




(race)+ = race car

(tie)+ = tie it

(tops)+ = top spot

Pal(aba) = a b a a b a




(race)+ = race car

(tie)+ = tie it

(tops)+ = top spot


Pal(abc) = a b a c a b a




(race)+ = race car

(tie)+ = tie it

(tops)+ = top spot



Pal(race) =rarcrarerarcrar




(race)+ = race car

(tie)+ = tie it

(tops)+ = top spot




L(5, 3) = aabaabab = aPal(aba)b




(race)+ = race car

(tie)+ = tie it

(tops)+ = top spot




L(5, 3) = aabaabab = aPal(aba)b

L(7, 4) = aabaabaabab = aPal(abaa)b



Sturmian words: Palindromicity

Theorem (de Luca 1997)

An infinite word s over {a, b} is a standard Sturmian word if and only ifthere exists an infinite word ∆ = x1x2x3 · · · over {a, b} (not of the formuaω or ubω) such that

s = limn→∞

Pal(x1x2 · · · xn) = Pal(∆).






s = limn→∞

Pal(x1x2 · · · xn) = Pal(∆).

∆: directive word of s.






s = limn→∞

Pal(x1x2 · · · xn) = Pal(∆).

∆: directive word of s.

Example: Fibonacci word is directed by ∆ = (ab)(ab)(ab) · · · .



Recall: Fibonacci word

a a

a a

a

a a

b

b

b

b

Line of slope√

5−12 −→ Fibonacci word




a a

a a

a

a a

b

b

b

b

Line of slope√


∆ = (ab)(ab)(ab) · · · −→ f = a




a a

a a

a

a a

b

b

b

b

Line of slope√


∆ = (ab)(ab)(ab) · · · −→ f = ab




a a

a a

a

a a

b

b

b

b

Line of slope√


∆ = (ab)(ab)(ab) · · · −→ f = aba




a a

a a

a

a a

b

b

b

b

Line of slope√


∆ = (ab)(ab)(ab) · · · −→ f = abaa




a a

a a

a

a a

b

b

b

b

Line of slope√


∆ = (ab)(ab)(ab) · · · −→ f = abaaba




a a

a a

a

a a

b

b

b

b

Line of slope√


∆ = (ab)(ab)(ab) · · · −→ f = abaabab




a a

a a

a

a a

b

b

b

b

Line of slope√


∆ = (ab)(ab)(ab) · · · −→ f = abaababaaba · · ·




a a

a a

a

a a

b

b

b

b

Line of slope√


∆ = (ab)(ab)(ab) · · · −→ f = abaababaaba · · ·

Note: Palindromic prefixes have lengths (Fn+1 − 2)n≥1 = 0, 1, 3, 6, 11, 19, . . . where(Fn)n≥0 is the sequence of Fibonacci numbers 1, 1, 2, 3, 5, 8, 13, 21, . . . ,defined by: F0 = F1 = 1,Fn = Fn−1 + Fn−2 for n ≥ 2.



A generalisation: Episturmian words

{a, b} −→ A (finite alphabet) gives standard episturmian words.

Theorem (Droubay-Justin-Pirillo 2001)

An infinite word s over A is a standard episturmian word if and only ifthere exists an infinite word ∆ = x1x2x3 · · · over A such that

s = limn→∞

Pal(x1x2 · · · xn) = Pal(∆).







s = limn→∞

Pal(x1x2 · · · xn) = Pal(∆).

Example: ∆ = (abc)(abc)(abc) · · · directs the Tribonacci word:







s = limn→∞

Pal(x1x2 · · · xn) = Pal(∆).


r = a







s = limn→∞

Pal(x1x2 · · · xn) = Pal(∆).


r = ab







s = limn→∞

Pal(x1x2 · · · xn) = Pal(∆).


r = aba







s = limn→∞

Pal(x1x2 · · · xn) = Pal(∆).


r = abac







s = limn→∞

Pal(x1x2 · · · xn) = Pal(∆).


r = abacaba







s = limn→∞

Pal(x1x2 · · · xn) = Pal(∆).


r = abacabaa







s = limn→∞

Pal(x1x2 · · · xn) = Pal(∆).


r = abacabaabacaba







s = limn→∞

Pal(x1x2 · · · xn) = Pal(∆).


r = abacabaabacabab







s = limn→∞

Pal(x1x2 · · · xn) = Pal(∆).


r = abacabaabacababacabaabacaba







s = limn→∞

Pal(x1x2 · · · xn) = Pal(∆).


r = abacabaabacababacabaabacabac







s = limn→∞

Pal(x1x2 · · · xn) = Pal(∆).


r = abacabaabacababacabaabacabacabaabaca · · ·







s = limn→∞

Pal(x1x2 · · · xn) = Pal(∆).


r = abacabaabacababacabaabacabacabaabaca · · ·Note: Palindromic prefixes have lengths ((Tn+2 + Tn + 1)/2 − 2)n≥1

= 0, 1, 3, 7, 14, 27, 36 . . . where (Tn)n≥0 is the sequence of Tribonaccinumbers 1, 1, 2, 4, 7, 13, 24, 44, . . . , defined by:

T0 = T1 = 1, T2 = 2, Tn = Tn−1 + Tn−2 + Tn−3 for n ≥ 3.



A further generalisation: Rich words

Sturmian and episturmian words are “rich” in palindromes.





Droubay, Justin, Pirillo (2001): Any finite word w of length |w |contains at most |w |+ 1 distinct palindromes (including ε).






G.-Justin (2007): Initiated a unified study of finite and infinite wordsthat are characterized by containing the maximal number of distinctpalindromes, called rich words.







A finite or infinite word w is rich if each factor u of w contains exactly|u|+ 1 distinct palindromic factors.








Characteristic property: All ‘complete returns’ to palindromes arepalindromes.









Examples: Sturmian and episturmian words









Examples: aaaaaa · · ·









Examples: abbbbbb · · ·









Examples: abaabaaabaaaab · · ·









Examples: (abcba)(abcba)(abcba) · · ·


Some Applications in Number Theory

Outline




Some Applications in Number Theory Continued Fractions & Sturmian Words

Continued fractions

Every irrational number α > 0 has a unique continued fraction expansion

α = [a0; a1, a2, a3, . . .] = a0 +1

a1 +1

a2 +1

a3 + · · ·where the ai are non-negative integers, called partial quotients, with a0 ≥ 0& all other ai ≥ 1. The n-th convergent to α is the rational number:

pn

qn

= [a0; a1, a2, . . . , an], n ≥ 1.



Continued fractions

Every irrational number α > 0 has a unique continued fraction expansion

α = [a0; a1, a2, a3, . . .] = a0 +1

a1 +1

a2 +1

a3 + · · ·where the ai are non-negative integers, called partial quotients, with a0 ≥ 0& all other ai ≥ 1. The n-th convergent to α is the rational number:

pn

qn

= [a0; a1, a2, . . . , an], n ≥ 1.

Example:

Golden ratio (conjugate): τ̄ = 1/τ =√

5−12 = 0.61803 . . . = [0; 1, 1, 1, . . .]

Convergents: 11 = 1,

1

1 + 11

= 12 , 2

3 , 35 , 5

8 , . . . , Fn−1

Fn, . . .



Standard Sturmian words via CF expansions

Let cα denote the standard Sturmian word corresponding to the line ofirrational slope α ∈ (0, 1) through (0, 0) where

α = [0; a1, a2, a3, . . .].





α = [0; a1, a2, a3, . . .].

Then cα = Pal(aa1ba2aa3ba4aa5 · · · ). [Arnoux-Rauzy 1991]





α = [0; a1, a2, a3, . . .].


Example: Recall the Fibonacci word is f := cα withα = 1/τ = (

√5− 1)/2 = [0; 1, 1, 1, 1, . . .] and combinatorially:

f = Pal(abababab · · · ) = abaababaaba · · · .





α = [0; a1, a2, a3, . . .].





cα can also be constructed as the limit of an infinite sequence of finitewords, defined with respect to the CF of α.





α = [0; a1, a2, a3, . . .].





cα can also be constructed as the limit of an infinite sequence of finitewords, defined with respect to the CF of α.

Many nice combinatorial properties of cα are related to the CF of α.


Some Applications in Number Theory Distribution modulo 1 & Sturmian Words

Fractional parts of powers & Sturmian words

Mahler (1968): defined the set of Z -numbers by

Z :=

{

ξ ∈ R, ξ > 0, ∀n ≥ 0, 0 ≤{

ξ

(3

2

)n}

<1

2

}

where {z} denotes the fractional part of z .





Z :=

{

ξ ∈ R, ξ > 0, ∀n ≥ 0, 0 ≤{

ξ

(3

2

)n}

<1

2

}


Mahler proved that Z is at most countable.





Z :=

{

ξ ∈ R, ξ > 0, ∀n ≥ 0, 0 ≤{

ξ

(3

2

)n}

<1

2

}



It is still an open problem to prove that Z = ∅.





Z :=

{

ξ ∈ R, ξ > 0, ∀n ≥ 0, 0 ≤{

ξ

(3

2

)n}

<1

2

}



It is still an open problem to prove that Z = ∅.

More general question:

Given a real number α > 1 and an interval (x , y) ⊂ (0, 1),does there exists ξ > 0 such that, for all n ≥ 0, we havex ≤ {ξαn} < y (or the variant x ≤ {ξαn} ≤ y)?



Fractional parts of powers & Sturmian words . . .

Flatto-Lagarias-Pollington, 1995

If p, q are coprime integers and p > q ≥ 2, then any interval (x , y) suchthat for some ξ > 0, one has that {ξ(p/q)n} ∈ (x , y) for all n ≥ 0, mustsatisfy y − x ≥ 1/p.






Bugeaud-Dubickas (2005): described all irrational numbers ξ > 0 such thatfor a fixed integer b ≥ 2 the fractional parts {ξbn}, n ≥ 0, all belong to aninterval of length 1/b.






Bugeaud-Dubickas (2005): described all irrational numbers ξ > 0 such thatfor a fixed integer b ≥ 2 the fractional parts {ξbn}, n ≥ 0, all belong to aninterval of length 1/b.

Bugeaud-Dubickas, 2005

Let b ≥ 2 be an integer and ξ > 0 be an irrational number. Then:

the numbers {ξbn} cannot all lie in an interval of length < 1/b;

there exists a closed interval of length 1/b containing the numbers{ξbn} for all n ≥ 0 iff the base b expansion of the fractional part of ξis a Sturmian sequence on {k , k + 1} for some k ∈ {0, 1, . . . , b− 2}.




The core of Bugeaud & Dubickas’ result is the following:

Theorem

An aperiodic sequence s := (sn)n≥0 on {0, 1} is Sturmian if and only ifthere exists a sequence u := (un)n≥0 on {0, 1} such that

0u ≤ T k(s) ≤ 1u for all k ≥ 0,

where T k denotes the k-th iterate of the shift map T defined byT ((sn)n≥1) = (sn+1)n≥1 and ≤ is the lexicographic order on sequencesover {0, 1} induced by 0 < 1.

Moreover, u is the unique standard Sturmian sequence with the same slopeas s , and we have

0u = inf{T k(s), k ≥ 0} and 1u = sup{T k(s), k ≥ 0}.




In particular, all shifts of a Sturmian sequence s of slope α over {0, 1}are lexicographically ≥ 0cα and lexicographically ≤ 1cα.





Example: Consider the Fibonacci word on {0, 1}:f = 0100101001001010010 · · · ,

the standard Sturmian word of slope α = (√

5− 1)/2.







5− 1)/2.

001001010010010 · · ·︸︷︷︸

0f

<

f︷︸︸︷

01001010010010 · · · < 101001010010010 · · ·︸︷︷︸

1f







5− 1)/2.

001001010010010 · · ·︸︷︷︸

0f

<

T (f )︷︸︸︷

1001010010010 · · · < 101001010010010 · · ·︸︷︷︸

1f







5− 1)/2.

001001010010010 · · ·︸︷︷︸

0f

<

T2(f )

︷︸︸︷

001010010010 · · · < 101001010010010 · · ·︸︷︷︸

1f







5− 1)/2.

001001010010010 · · ·︸︷︷︸

0f

<

T3(f )

︷︸︸︷

01010010010 · · · < 101001010010010 · · ·︸︷︷︸

1f







5− 1)/2.

001001010010010 · · ·︸︷︷︸

0f

<

T4(f )

︷︸︸︷

1010010010 · · · < 101001010010010 · · ·︸︷︷︸

1f

and so on . . .







5− 1)/2.

001001010010010 · · ·︸︷︷︸

0f

<

T4(f )

︷︸︸︷

1010010010 · · · < 101001010010010 · · ·︸︷︷︸

1f

and so on . . .

The preceding theorem has been rediscovered numerous times sincethe work of P. Veerman in the mid-late 80’s (which was the first timeit was proved as far as we can ascertain).



The main tool used by Bugeaud and Dubickas was combinatorics on

words: replace real numbers by their base b expansions, and transforminequalities between real numbers into (lexicographic) inequalitiesbetween infinite sequences representing their base b expansions.





Given a sequence s over {0, 1}, let r(s) denote the real number whosebase 2 expansion is s .

Now let x , y , s be sequences over {0, 1}. Then

x ≤ T k(s) ≤ y ←→ r(x) ≤ {r(s)2k} ≤ r(y).







x ≤ T k(s) ≤ y ←→ r(x) ≤ {r(s)2k} ≤ r(y).

Example:

(01)∞ ≤ T k((01)∞) ≤ (10)∞ for all k ≥ 0

13 ≤ {1

3 · 2k} ≤ 23 for all k ≥ 0







x ≤ T k(s) ≤ y ←→ r(x) ≤ {r(s)2k} ≤ r(y).

Example:

(01)∞ ≤ T k((01)∞) ≤ (10)∞ for all k ≥ 0

13 ≤ {1

3 · 2k} ≤ 23 for all k ≥ 0

We have obtained a complete description of the minimal intervalscontaining all fractional parts {ξ2n}, n ≥ 0, for some real number ξ > 0 . . .Amy Glen (Murdoch University) CoW & Number Theory November 17, 2009 42 / 51


Definition

For all x ∈ [0, 1], let Sx := {ξ ∈ R, ξ > 0, ∀n ≥ 0, x ≤ {ξ2n} < 1} andlet F : [0, 1]→ [0, 1] be the function defined by:

F (x) =

{

inf{y ∈ [0, 1), ∃ξ > 0, ∀n ≥ 0, x ≤ {ξ2n} ≤ y} if Sx 6= ∅,1 if Sx = ∅.



Definition


F (x) =

{


From Bugeaud & Dubickas’ result for b = 2, we deduce the following facts.

For x ∈ [12 , 1], there does not exist a real number ξ > 0 such thatx ≤ {ξ2n} < 1 for all n ≥ 0.



Definition


F (x) =

{




Hence, F (x) = 1 for all x ∈ [12 , 1].



Definition


F (x) =

{




Hence, F (x) = 1 for all x ∈ [12 , 1].

If ξ > 0 is an irrational real number, then there exists a real numberx ∈ [0, 1

2 ) such that all the fractional parts {ξ2n} belong to theinterval [x , x + 1

2 ] if and only if the base 2 expansion of the fractionalpart of ξ is a Sturmian sequence.



Definition


F (x) =

{




Hence, F (x) = 1 for all x ∈ [12 , 1].

If ξ > 0 is an irrational real number, then there exists a real numberx ∈ [0, 1

2 ) such that all the fractional parts {ξ2n} belong to theinterval [x , x + 1

2 ] if and only if the base 2 expansion of the fractionalpart of ξ is a Sturmian sequence.

Furthermore, for any such x , one has F (x) = x + 12 .



Theorem (Allouche-G., 2009)

Let x be a real number in [0, 1].

(i) If x ≥ 12 , then F (x) = 1.





(i) If x ≥ 12 , then F (x) = 1.

(ii) If x = 0, then F (x) = 0.





(i) If x ≥ 12 , then F (x) = 1.

(ii) If x = 0, then F (x) = 0.

(iii) If x ∈ (0, 12) and if the base 2 expansion of 2x is given by a standard

Sturmian sequence, then F (x) = x + 12 .

Furthermore, F (x) is the unique real number in [0, 1] that has aSturmian base 2 expansion and satisfies x ≤ {F (x)2k} ≤ F (x) for allk ≥ 0.





(i) If x ≥ 12 , then F (x) = 1.

(ii) If x = 0, then F (x) = 0.

(iii) If x ∈ (0, 12) and if the base 2 expansion of 2x is given by a standard

Sturmian sequence, then F (x) = x + 12 .

Furthermore, F (x) is the unique real number in [0, 1] that has aSturmian base 2 expansion and satisfies x ≤ {F (x)2k} ≤ F (x) for allk ≥ 0.

Let Sα denote the set of all Sturmian sequences of (irrational) slopeα ∈ (0, 1) over the alphabet {0, 1} (i.e., a 7→ 0, b 7→ 1).

Noting that r(1cα) = 1/2 + r(0cα), we have that the setr(Sα) := {r(s) ∈ [0, 1), s ∈ Sα} is completely contained within theclosed interval [r(0cα), r(1cα)] of length 1/2 (and no smaller).



Theorem (Allouche-G., 2009) . . .

(iv) If x ∈ (0, 12) and if the base 2 expansion of 2x is a periodic sequence

of the form (Pal(v)01)∞ or (Pal(v)10)∞ for some v ∈ {0, 1}∗, thenF (x) is the rational number whose base 2 expansion is the periodicsequence (1Pal(v)0)∞, in which case F (x) ≤ x + 1

2 .






2 .

(v) In all other cases, F (x) can be explicitly computed: it is equal to therational number whose base 2 expansion is a periodic sequence of theform (1Pal(v)0)∞ for some (unique) v ∈ {0, 1}∗ such that

r((Pal(v)01)∞) < 2x < r((Pal(v)10)∞).

In these cases, F (x) < x + 12 ·






2 .

(v) In all other cases, F (x) can be explicitly computed: it is equal to therational number whose base 2 expansion is a periodic sequence of theform (1Pal(v)0)∞ for some (unique) v ∈ {0, 1}∗ such that

r((Pal(v)01)∞) < 2x < r((Pal(v)10)∞).

In these cases, F (x) < x + 12 ·

Moreover, in cases (iv) and (v), F (x) is the unique real number in (0, 1)whose base 2 expansion is a periodic sequence of the form (1Pal(v)0)∞

and which satisfies x ≤ {F (x)2k} ≤ F (x) for all k ≥ 0.Amy Glen (Murdoch University) CoW & Number Theory November 17, 2009 45 / 51


Examples

F (1/4) = 2/3 since the base 2 expansion of 1/4 is

(1/4)2 = 01000 · · · (or 00111 · · · ),and we have

(01)∞ < (2× (1/4))2 (= 1000 · · · ) < (10)∞

where (10)∞ is the base 2 expansion of 2/3. (Here Pal(v) = ε.)



Examples

F (1/4) = 2/3 since the base 2 expansion of 1/4 is

(1/4)2 = 01000 · · · (or 00111 · · · ),and we have

(01)∞ < (2× (1/4))2 (= 1000 · · · ) < (10)∞

where (10)∞ is the base 2 expansion of 2/3. (Here Pal(v) = ε.)

F (1/2π) = 20/31 (< 1/2π + 1/2 = 0.65915 . . .) since the base 2expansion of 1/2π is

(1/2π)2 = 0 01010︸︷︷︸

Pal(011)

00101111100110000011011011100 · · ·

and we have

(01001)∞ < (2× (1/2π))2 < (01010)∞

where (10100)∞ is the base 2 expansion of 20/31.



Remarks

It is known (Ferenczi-Mauduit, 1997) that real numbers having aSturmian base 2 expansion are transcendental.

As a consequence of our theorem, we deduce:

If x is an algebraic real number in [0, 12), then F (x) is rational.



Remarks




One may ask what happens with base b expansions, where b ≥ 3, orwhat can be said about the intervals containing all {ξbn} for some ξ.



Remarks




One may ask what happens with base b expansions, where b ≥ 3, orwhat can be said about the intervals containing all {ξbn} for some ξ.

The result of Bugeaud and Dubickas (2005) implies that Sturmiansequences on an alphabet {k , k + 1} for some k ∈ {0, 1, . . . , b − 2}will again play a fundamental role.


Some Applications in Number Theory Transcendental Numbers

Transcendental continued fractions

Long-standing Conjecture (Khintchine 1949)

The CF expansion of an irrational algebraic real number α is eithereventually periodic (iff α is a quadratic irrational) or it contains arbitrarilylarge partial quotients.






Alternatively: An irrational number whose CF expansion has boundedpartial quotients is either quadratic or transcendental.







Liouville (1844): Transcendental CF’s whose sequences of partialquotients grow very fast (too fast to be algebraic)







Liouville (1844): Transcendental CF’s whose sequences of partialquotients grow very fast (too fast to be algebraic)

Transcendental CF’s with bounded partial quotients:

Maillet (1906)Baker (1962, 1964)Shallit (1979)Davison (1989)Queffélec (1998)Allouche, Davison, Queffélec, Zamboni (2001)Adamczewski-Bugeaud (2005)

9

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;

transcendence criteria from DA



Transcendental continued fractions: Examples

Let ξa,b := [0; a, b, a, a, b, a, b, a, a, b, a, . . .] where the sequence ofpartial quotients is the Fibonacci word on positive integers a, b.





Any Fibonacci continued fraction is transcendental.






More generally:

Theorem (Allouche-Davison-Queffélec-Zamboni 2001)

Any Sturmian continued fraction – of which the partial quotients forms aSturmian word on two distinct positive integers – is transcendental.






More generally:



Proved by showing that Sturmian continued fractions admit very goodapproximations by quadratic real numbers.






More generally:




In particular, they used the fact that any Sturmian word begins witharbitrarily long squares (words of the form XX = X 2).






More generally:





Example: f = aba · ababaabaababaababaabaababaaba · · ·Amy Glen (Murdoch University) CoW & Number Theory November 17, 2009 49 / 51





More generally:





Example: f = abaab · abaabaababaababaabaababaaba · · ·Amy Glen (Murdoch University) CoW & Number Theory November 17, 2009 49 / 51





More generally:





Example: f = abaababa · abaababaababaabaababaaba · · ·Amy Glen (Murdoch University) CoW & Number Theory November 17, 2009 49 / 51





More generally:





Example: f = abaababaabaab · abaababaabaababaaba · · ·Amy Glen (Murdoch University) CoW & Number Theory November 17, 2009 49 / 51


Transcendental continued fractions . . .

In fact: Any irrational number having a CF expansion with sequence ofpartial quotients forming a recurrent rich infinite word is either quadratic ortranscendental by . . .





Theorem (Adamczewski-Bugeaud 2007)

If the sequence of partial quotients (an)n≥0 in the CF expansion of apositive irrational number ξ := [a0; a1, a2, . . . , an, . . .] begins with arbitrarilylong palindromes, then ξ is either quadratic or transcendental.







Palindromes must begin at a0.







Palindromes must begin at a0.

Proof of theorem rests on Schmidt’s Subspace Theorem (1972).


Thank You!


combinatorics on words & some applications in number theory

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