Palindromic sequences in Number Theory - · PDF filePalindromic sequences in Number Theory Amy Glen The Mathematics Institute @ Reykjavík University [email protected]

Palindromic sequences in Number Theory

Amy Glen

The Mathematics Institute @ Reykjavík University

[email protected]://www.ru.is/kennarar/amy

Department of Mathematics and Statistics@

University of Winnipeg

Amy Glen (Reykjavík University) Palindromes in Number Theory April 2009 1 / 48

Outline

1 Combinatorics on WordsSturmian & Episturmian Words

2 Some Connections to Number TheoryContinued Fractions & Sturmian WordsPalindromes & Diophantine ApproximationTranscendental NumbersMiscellaneous


Combinatorics on Words

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 (Aussie 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 = bpa

where 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

(ab)+ =




(race)+ = race car

(tie)+ = tie it

(tops)+ = top spot

(ab)+ = aba




(race)+ = race car

(tie)+ = tie it

(tops)+ = top spot

(ab)+ = aba

Pal(aba) =




(race)+ = race car

(tie)+ = tie it

(tops)+ = top spot

(ab)+ = aba

Pal(aba) = a




(race)+ = race car

(tie)+ = tie it

(tops)+ = top spot

(ab)+ = aba

Pal(aba) = a b




(race)+ = race car

(tie)+ = tie it

(tops)+ = top spot

(ab)+ = aba

Pal(aba) = a b a




(race)+ = race car

(tie)+ = tie it

(tops)+ = top spot

(ab)+ = aba

Pal(aba) = a b a a




(race)+ = race car

(tie)+ = tie it

(tops)+ = top spot

(ab)+ = aba

Pal(aba) = a b a a b a




(race)+ = race car

(tie)+ = tie it

(tops)+ = top spot

(ab)+ = aba


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




(race)+ = race car

(tie)+ = tie it

(tops)+ = top spot

(ab)+ = aba



Pal(race) =




(race)+ = race car

(tie)+ = tie it

(tops)+ = top spot

(ab)+ = aba



Pal(race) = rarcrarerarcrar




(race)+ = race car

(tie)+ = tie it

(tops)+ = top spot

(ab)+ = aba




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




(race)+ = race car

(tie)+ = tie it

(tops)+ = top spot

(ab)+ = aba




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 Tribonacci

numbers 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.


Some Connections to Number Theory

Outline




Some Connections to 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


α = [a0; a1, a2, a3, . . .] = a0 +1

a1 +1

a2 +1


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, . . .



Continued fractions


α = [a0; a1, a2, a3, . . .] = a0 +1

a1 +1

a2 +1


pn

qn

= [a0; a1, a2, . . . , an], n ≥ 1.

Example:

π = [3, 7, 15, 1, 292, 1, 1, 1, 2, 1, 3, 1, 14, 2, . . .]

Convergents: 3, 22/7, 333/106, 355/113, . . .

Note: 2[1, 1, 1, 3, 32] = 355/113 = 3.14159292 ≈ π . . . → v. good approx.



Standard Sturmian words: CF-based construction

Suppose α = [0; d1, d2, d3, . . .].

To the directive sequence (d1, d2, d3, . . .), we associate a sequence(sn)n≥−1 of words defined by

s−1 = b, s0 = a, sn = sdn

n−1sn−2, n ≥ 1.

For all n ≥ 1, sn−1 is a prefix of sn and the standard Sturmian word cα

corresponding to the line of slope α through (0, 0) is given by

cα = limn→∞ sn.




Suppose α = [0; d1, d2, d3, . . .].


s−1 = b, s0 = a, sn = sdn

n−1sn−2, n ≥ 1.




Example: Recall the Fibonacci word is f := cα with α = [0; 1, 1, 1, 1, . . .].

s−1 = b, s0 = a, and sn = sn−1sn−2 for n ≥ 1 −→ |sn| = Fn+1




Suppose α = [0; d1, d2, d3, . . .].


s−1 = b, s0 = a, sn = sdn

n−1sn−2, n ≥ 1.






f = ab s1, length F2 = 2




Suppose α = [0; d1, d2, d3, . . .].


s−1 = b, s0 = a, sn = sdn

n−1sn−2, n ≥ 1.






f = aba s2, length F3 = 3




Suppose α = [0; d1, d2, d3, . . .].


s−1 = b, s0 = a, sn = sdn

n−1sn−2, n ≥ 1.






f = abaab s3, length F4 = 5




Suppose α = [0; d1, d2, d3, . . .].


s−1 = b, s0 = a, sn = sdn

n−1sn−2, n ≥ 1.






f = abaababa s4, length F5 = 8




Suppose α = [0; d1, d2, d3, . . .].


s−1 = b, s0 = a, sn = sdn

n−1sn−2, n ≥ 1.






f = abaababaabaab s5, length F6 = 13




Suppose α = [0; d1, d2, d3, . . .].


s−1 = b, s0 = a, sn = sdn

n−1sn−2, n ≥ 1.






f = abaababaabaababaababa s6, length F7 = 21




Suppose α = [0; d1, d2, d3, . . .].


s−1 = b, s0 = a, sn = sdn

n−1sn−2, n ≥ 1.






f = abaababaabaababaababa · · ·




Suppose α = [0; d1, d2, d3, . . .].


s−1 = b, s0 = a, sn = sdn

n−1sn−2, n ≥ 1.




In general:

|sn| = qn+1 for all n




Suppose α = [0; d1, d2, d3, . . .].


s−1 = b, s0 = a, sn = sdn

n−1sn−2, n ≥ 1.




In general:


sn = Pal(v)xy for some v ∈ {a, b}∗ and {x , y} = {a, b}




Suppose α = [0; d1, d2, d3, . . .].


s−1 = b, s0 = a, sn = sdn

n−1sn−2, n ≥ 1.




In general:


sn = Pal(v)xy for some v ∈ {a, b}∗ and {x , y} = {a, b}cα = Pal(ad1bd2ad3bd4 · · · ) E.g. f = Pal(abababa . . .)




Suppose α = [0; d1, d2, d3, . . .].


s−1 = b, s0 = a, sn = sdn

n−1sn−2, n ≥ 1.




In general:


sn = Pal(v)xy for some v ∈ {a, b}∗ and {x , y} = {a, b}cα = Pal(ad1bd2ad3bd4 · · · ) E.g. f = Pal(abababa . . .)

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


Some Connections to Number Theory Palindromes & Diophantine Approximation

Basics of Diophantine Approximation

Diophantine Approximation is concerned with the approximation ofreal numbers by rational numbers.





How good is an approximation of a real number ξ by a rationalnumber p/q?






Rudimentary measure: |ξ − p/q| → the smaller the better.







A more sophisticated measure: Comparison of |ξ − p/q| to the size of q

(in terms of some function φ(q)).









Typically, it is asked whether or not an inequality of the form|ξ − p/q| < φ(q) has infinitely many rational solutions p/q.










Relation to continued fractions:

Best rational approximations to real numbers are produced bytruncating their CF expansions.










Relation to continued fractions:

Best rational approximations to real numbers are produced bytruncating their CF expansions.

CF theory → for every irrational number ξ, the inequality|ξ − p/q| < 1/q2 always has infinitely many rational solutions p/q.



Palindromes & Diophantine Approximation

Roy 2003, 2004: Introduced a “palindromic prefix method” in thestudy of simultaneous approximation to a real number and its square.





Roy’s work inspired Fischler (2006) to study infinite words with“abundant palindromic prefixes”.






Also of interest in Physics in connection with the spectral theory ofone-dimensional Schrödinger operators. [Hof, Knill, Simon 1995]







Important notion is palindromic complexity – the number of distinctpalindromic factors of each length.







Important notion is palindromic complexity – the number of distinctpalindromic factors of each length.

Fischler introduced palindromic prefix density . . .



Palindromic prefix density

Let w = w1w2w3 · · · be an infinite word (with each wi a letter).





Let (ni )i≥1 be the seq. of lengths of the palindromic prefixes of w.

Note: (ni )i≥1 is an increasing sequence if w begins with arbitrarilylong palindromes.







Fischler (2006): The palindromic prefix density of w is defined by

dp(w) :=

(

lim supi→∞

ni+1

ni

)−1

with dp(w) := 0 if w begins with only finitely many palindromes.








dp(w) :=

(

lim supi→∞

ni+1

ni

)−1


Note: 0 ≤ dp(w) ≤ 1 for any infinite word w.








dp(w) :=

(

lim supi→∞

ni+1

ni

)−1


Note: 0 ≤ dp(w) ≤ 1 for any infinite word w.

If w = v∞ = vvvvv · · · (purely periodic), then

dp(w) =

{1 if v = pq for some palindromes p, q

0 otherwise.



Palindromic prefix density . . .

Question: What is the maximal palindromic prefix density attainable by anon-periodic infinite word?

Answer:

Theorem (Fischler 2006)

For any infinite non-periodic word w, we have dp(w) ≤ 1τ (= (

√5 − 1)/2).





Answer:



√5 − 1)/2).

Fischler’s bound is optimal . . .





Answer:



√5 − 1)/2).

Fischler’s bound is optimal . . .

The Fibonacci word f = abaababaaba · · · , whose sequence of palindromicprefix lengths is given by:

(ni )i≥1 = (Fi+1 − 2)i≥1 = 0, 1, 3, 6, 11, 19, 32, . . . ,

has maximal palindromic prefix density amongst non-periodic infinitewords. That is:

dp(f) =

(

lim supi→∞

Fi+2 − 2

Fi+1 − 2

)−1

= 1/τ.



Palindrome prefix density . . .

There is an easy formula to compute dp(cα). [de Luca 1997]

Not so for standard episturmian words. But it can be verified that . . .






Any standard episturmian word satisfies ni+1 ≤ 2ni + 1 for any i .

Recall: The Tribonacci word r = abacabaabacababacab · · · haspalindromic prefix lengths

(ni )i≥1 =

(Ti+2 + Ti + 1

2− 2

)

i≥1

= 0, 1, 3, 7, 14, 27, 36 . . .






Any standard episturmian word satisfies ni+1 ≤ 2ni + 1 for any i .

Recall: The Tribonacci word r = abacabaabacababacab · · · haspalindromic prefix lengths

(ni )i≥1 =

(Ti+2 + Ti + 1

2− 2

)

i≥1

= 0, 1, 3, 7, 14, 27, 36 . . .

Fischler 2006: Introduced a natural generalisation of standardepisturmian words . . .



Fischler’s words with abundant palindromic prefixes

Definition (Fischler 2006)

An infinite word w is said to have abundant palindromic prefixes if thesequence (ni )i≥1 is infinite and satisfies ni+1 ≤ 2ni + 1 for any i ≥ 1.






Fischler gave an explicit construction of such words, as well as thosewhich satisfy ni+1 ≤ 2ni + 1 for sufficiently large i .







Construction is similar to that of iterated palindromic closure Pal forstandard episturmian words.








Any standard episturmian word has abundant palindromic prefixes, butnot conversely.








Any standard episturmian word has abundant palindromic prefixes, butnot conversely.

Example: (abcacba)∞ = abcacbaabcacbaabcacba · · · has abundantpalindromic prefixes, but it is not standard episturmian.



A further generalisation: Rich words

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





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






Examples

aaaaaa · · ·






Examples

aaaaaa · · ·abbbbbb · · ·






Examples

aaaaaa · · ·abbbbbb · · ·abaabaaabaaaab · · ·






Examples

aaaaaa · · ·abbbbbb · · ·abaabaaabaaaab · · ·(abcba)(abcba)(abcba) · · ·






Examples

aaaaaa · · ·abbbbbb · · ·abaabaaabaaaab · · ·(abcba)(abcba)(abcba) · · ·Sturmian and episturmian words






Examples

aaaaaa · · ·abbbbbb · · ·abaabaaabaaaab · · ·(abcba)(abcba)(abcba) · · ·Sturmian and episturmian words

Infinite words with abundant palindromic prefixes


Some Connections to 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)

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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 (Reykjavík University) Palindromes in Number Theory April 2009 45 / 48





More generally:





Example: f = abaab · abaabaababaababaabaababaaba · · ·Amy Glen (Reykjavík University) Palindromes in Number Theory April 2009 45 / 48





More generally:





Example: f = abaababa · abaababaababaabaababaaba · · ·Amy Glen (Reykjavík University) Palindromes in Number Theory April 2009 45 / 48





More generally:





Example: f = abaababaabaab · abaababaabaababaaba · · ·Amy Glen (Reykjavík University) Palindromes in Number Theory April 2009 45 / 48


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).


Some Connections to Number Theory Miscellaneous

Further work

Continued fractions provide a strong link between:

arithmetic/Diophantine properties of an irrational number α,

and symbolic/combinatorial properties of Sturmian words of slope α.



Further work




Strict episturmian words (or Arnoux-Rauzy sequences) naturallygeneralise and extend this rich interplay. [Rauzy 1982, Arnoux-Rauzy 1991]



Further work





Recall: Directive word of cα is determined by the CF of α.

That is: If α = [0; d1, d2, d3, d4 . . .], then cα = Pal(ad1bd2ad3bd4 · · · ).



Further work






That is: If α = [0; d1, d2, d3, d4 . . .], then cα = Pal(ad1bd2ad3bd4 · · · ).Likewise, the directive word of a k-letter Arnoux-Rauzy word isdetermined by a multi-dimensional continued fraction expansion of thefrequencies of the first k − 1 letters. [Zamboni 1998, Wozny-Zamboni 2001]



Further work






That is: If α = [0; d1, d2, d3, d4 . . .], then cα = Pal(ad1bd2ad3bd4 · · · ).Likewise, the directive word of a k-letter Arnoux-Rauzy word isdetermined by a multi-dimensional continued fraction expansion of thefrequencies of the first k − 1 letters. [Zamboni 1998, Wozny-Zamboni 2001]

Deep properties studied in the framework of dynamical systems, withconnections to geometrical realisations such as Rauzy fractals.



Thank you!


Documents

Palindromic sequences in Number Theory - · PDF filePalindromic sequences in Number Theory Amy Glen The Mathematics Institute @ Reykjavík University [email protected]