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
Amy Glen (Reykjavík University) Palindromes in Number Theory April 2009 2 / 48
Combinatorics on Words
Outline
1 Combinatorics on WordsSturmian & Episturmian Words
2 Some Connections to Number TheoryContinued Fractions & Sturmian WordsPalindromes & Diophantine ApproximationTranscendental NumbersMiscellaneous
Amy Glen (Reykjavík University) Palindromes in Number Theory April 2009 3 / 48
Combinatorics on Words
Starting point: Combinatorics on words
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
Amy Glen (Reykjavík University) Palindromes in Number Theory April 2009 4 / 48
Combinatorics on Words
Starting point: Combinatorics on words
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
A word w is a finite or infinite sequence of symbols (letters) taken from anon-empty finite set A (alphabet).
Amy Glen (Reykjavík University) Palindromes in Number Theory April 2009 4 / 48
Combinatorics on Words
Starting point: Combinatorics on words
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
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 · · ·
Amy Glen (Reykjavík University) Palindromes in Number Theory April 2009 4 / 48
Combinatorics on Words
Combinatorics on words: A brief history
Relatively new area of Discrete Mathematics
Amy Glen (Reykjavík University) Palindromes in Number Theory April 2009 5 / 48
Combinatorics on Words
Combinatorics on words: A brief history
Relatively new area of Discrete Mathematics
Early 1900’s: First investigations by Axel Thue (repetitions in words)
Amy Glen (Reykjavík University) Palindromes in Number Theory April 2009 5 / 48
Combinatorics on Words
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.
Amy Glen (Reykjavík University) Palindromes in Number Theory April 2009 5 / 48
Combinatorics on Words
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.
Amy Glen (Reykjavík University) Palindromes in Number Theory April 2009 5 / 48
Combinatorics on Words
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.
1960’s: Systematic study initiated by M.P. Schützenberger.
Amy Glen (Reykjavík University) Palindromes in Number Theory April 2009 5 / 48
Combinatorics on Words
Combinatorics on words: Complexity
Most commonly studied words are those which satisfy one or morestrong regularity properties; for instance, words containing manyrepetitions or palindromes.
Amy Glen (Reykjavík University) Palindromes in Number Theory April 2009 6 / 48
Combinatorics on Words
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).
Amy Glen (Reykjavík University) Palindromes in Number Theory April 2009 6 / 48
Combinatorics on Words
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”.
Amy Glen (Reykjavík University) Palindromes in Number Theory April 2009 6 / 48
Combinatorics on Words
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.
Amy Glen (Reykjavík University) Palindromes in Number Theory April 2009 6 / 48
Combinatorics on Words
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.
Example: w = abca has 9 distinct factors:
a, b, c ,︸ ︷︷ ︸
1
ab, bc , ca,︸ ︷︷ ︸
2
abc , bca,︸ ︷︷ ︸
3
abca.︸ ︷︷ ︸
4
Amy Glen (Reykjavík University) Palindromes in Number Theory April 2009 6 / 48
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.
Amy Glen (Reykjavík University) Palindromes in Number Theory April 2009 7 / 48
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
Aperiodic infinite words of minimal complexity – exactly n + 1 distinctfactors of length n for each n.
Amy Glen (Reykjavík University) Palindromes in Number Theory April 2009 7 / 48
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
Aperiodic infinite words of minimal complexity – exactly n + 1 distinctfactors of length n for each n.
Pioneering work by Morse & Hedlund in 1940 (symbolic dynamics).
Amy Glen (Reykjavík University) Palindromes in Number Theory April 2009 7 / 48
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
Aperiodic infinite words of minimal complexity – exactly n + 1 distinctfactors of length n for each n.
Pioneering work by Morse & Hedlund in 1940 (symbolic dynamics).
Low complexity accounts for many interesting features, as it inducescertain regularities, without periodicity.
Amy Glen (Reykjavík University) Palindromes in Number Theory April 2009 7 / 48
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
Aperiodic infinite words of minimal complexity – exactly n + 1 distinctfactors of length n for each n.
Pioneering work by Morse & Hedlund in 1940 (symbolic dynamics).
Low complexity accounts for many interesting features, as it inducescertain regularities, without periodicity.
Points of view: combinatorial; algebraic; geometric.
Amy Glen (Reykjavík University) Palindromes in Number Theory April 2009 7 / 48
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
Aperiodic infinite words of minimal complexity – exactly n + 1 distinctfactors of length n for each n.
Pioneering work by Morse & Hedlund in 1940 (symbolic dynamics).
Low complexity accounts for many interesting features, as it inducescertain regularities, without periodicity.
Points of view: combinatorial; algebraic; geometric.
References in: Combinatorics, Symbolic Dynamics, Number Theory,Discrete Geometry, Theoretical Physics, Theoretical Computer Science.
Amy Glen (Reykjavík University) Palindromes in Number Theory April 2009 7 / 48
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
Aperiodic infinite words of minimal complexity – exactly n + 1 distinctfactors of length n for each n.
Pioneering work by Morse & Hedlund in 1940 (symbolic dynamics).
Low complexity accounts for many interesting features, as it inducescertain regularities, without periodicity.
Points of view: combinatorial; algebraic; geometric.
References in: Combinatorics, Symbolic Dynamics, Number Theory,Discrete Geometry, Theoretical Physics, Theoretical Computer Science.
Numerous equivalent definitions & characterisations . . .
Amy Glen (Reykjavík University) Palindromes in Number Theory April 2009 7 / 48
Combinatorics on Words Sturmian & Episturmian Words
Christoffel words: A special family of finite Sturmian words
Lower Christoffel word of slope 35
Amy Glen (Reykjavík University) Palindromes in Number Theory April 2009 8 / 48
Combinatorics on Words Sturmian & Episturmian Words
Christoffel words: Construction
Lower Christoffel word of slope 35
Amy Glen (Reykjavík University) Palindromes in Number Theory April 2009 9 / 48
Combinatorics on Words Sturmian & Episturmian Words
Christoffel words: Construction
Lower Christoffel word of slope 35
Amy Glen (Reykjavík University) Palindromes in Number Theory April 2009 10 / 48
Combinatorics on Words Sturmian & Episturmian Words
Christoffel words: Construction
Lower Christoffel word of slope 35
Amy Glen (Reykjavík University) Palindromes in Number Theory April 2009 11 / 48
Combinatorics on Words Sturmian & Episturmian Words
Christoffel words: Construction
Lower Christoffel word of slope 35
Amy Glen (Reykjavík University) Palindromes in Number Theory April 2009 12 / 48
Combinatorics on Words Sturmian & Episturmian Words
Christoffel words: Construction
Lower Christoffel word of slope 35
Amy Glen (Reykjavík University) Palindromes in Number Theory April 2009 13 / 48
Combinatorics on Words Sturmian & Episturmian Words
Christoffel words: Construction
Lower Christoffel word of slope 35
Amy Glen (Reykjavík University) Palindromes in Number Theory April 2009 14 / 48
Combinatorics on Words Sturmian & Episturmian Words
Christoffel words: Construction
Lower Christoffel word of slope 35
Amy Glen (Reykjavík University) Palindromes in Number Theory April 2009 15 / 48
Combinatorics on Words Sturmian & Episturmian Words
Christoffel words: Construction
Lower Christoffel word of slope 35
Amy Glen (Reykjavík University) Palindromes in Number Theory April 2009 16 / 48
Combinatorics on Words Sturmian & Episturmian Words
Christoffel words: Construction
Lower Christoffel word of slope 35
a
L(5,3) = a
Amy Glen (Reykjavík University) Palindromes in Number Theory April 2009 17 / 48
Combinatorics on Words Sturmian & Episturmian Words
Christoffel words: Construction
Lower Christoffel word of slope 35
a a
L(5,3) = aa
Amy Glen (Reykjavík University) Palindromes in Number Theory April 2009 18 / 48
Combinatorics on Words Sturmian & Episturmian Words
Christoffel words: Construction
Lower Christoffel word of slope 35
a a
b
L(5,3) = aab
Amy Glen (Reykjavík University) Palindromes in Number Theory April 2009 19 / 48
Combinatorics on Words Sturmian & Episturmian Words
Christoffel words: Construction
Lower Christoffel word of slope 35
a a
ba
L(5,3) = aaba
Amy Glen (Reykjavík University) Palindromes in Number Theory April 2009 20 / 48
Combinatorics on Words Sturmian & Episturmian Words
Christoffel words: Construction
Lower Christoffel word of slope 35
a a
ba a
L(5,3) = aabaa
Amy Glen (Reykjavík University) Palindromes in Number Theory April 2009 21 / 48
Combinatorics on Words Sturmian & Episturmian Words
Christoffel words: Construction
Lower Christoffel word of slope 35
a a
ba a
b
L(5,3) = aabaab
Amy Glen (Reykjavík University) Palindromes in Number Theory April 2009 22 / 48
Combinatorics on Words Sturmian & Episturmian Words
Christoffel words: Construction
Lower Christoffel word of slope 35
a a
ba a
ba
L(5,3) = aabaaba
Amy Glen (Reykjavík University) Palindromes in Number Theory April 2009 23 / 48
Combinatorics on Words Sturmian & Episturmian Words
Christoffel words: Construction
Lower Christoffel word of slope 35
a a
ba a
ba
b
L(5,3) = aabaabab
Amy Glen (Reykjavík University) Palindromes in Number Theory April 2009 24 / 48
Combinatorics on Words Sturmian & Episturmian Words
Christoffel words: Construction
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
Amy Glen (Reykjavík University) Palindromes in Number Theory April 2009 25 / 48
Combinatorics on Words Sturmian & Episturmian Words
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.
Amy Glen (Reykjavík University) Palindromes in Number Theory April 2009 26 / 48
Combinatorics on Words Sturmian & Episturmian Words
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
Amy Glen (Reykjavík University) Palindromes in Number Theory April 2009 26 / 48
Combinatorics on Words Sturmian & Episturmian Words
Christoffel words: Historical notes
Before the 20th century:
J. Bernoulli, 1771 (Astronomy)
A. Markoff, 1882 (continued fractions)
E. Christoffel, 1871, 1888 (Cayley graphs)
Amy Glen (Reykjavík University) Palindromes in Number Theory April 2009 27 / 48
Combinatorics on Words Sturmian & Episturmian Words
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
Amy Glen (Reykjavík University) Palindromes in Number Theory April 2009 27 / 48
Combinatorics on Words Sturmian & Episturmian Words
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
Amy Glen (Reykjavík University) Palindromes in Number Theory April 2009 28 / 48
Combinatorics on Words Sturmian & Episturmian Words
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
Properties
L(p, q) = awb ⇐⇒ U(p, q) = bwa
Amy Glen (Reykjavík University) Palindromes in Number Theory April 2009 28 / 48
Combinatorics on Words Sturmian & Episturmian Words
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
Properties
L(p, q) = awb ⇐⇒ U(p, q) = bwa
|L(p, q)|a = p, |L(p, q)|b = q =⇒ |L(p, q)| = p + q
Amy Glen (Reykjavík University) Palindromes in Number Theory April 2009 28 / 48
Combinatorics on Words Sturmian & Episturmian Words
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
Properties
L(p, q) = awb ⇐⇒ U(p, q) = bwa
|L(p, q)|a = p, |L(p, q)|b = q =⇒ |L(p, q)| = p + q
L(p, q) is the reversal of U(p, q)
Amy Glen (Reykjavík University) Palindromes in Number Theory April 2009 28 / 48
Combinatorics on Words Sturmian & Episturmian Words
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
Properties
L(p, q) = awb ⇐⇒ U(p, q) = bwa
|L(p, q)|a = p, |L(p, q)|b = q =⇒ |L(p, q)| = p + q
L(p, q) is the reversal of U(p, q)
Christoffel words are of the form awb, bwa where w is a palindrome.
Amy Glen (Reykjavík University) Palindromes in Number Theory April 2009 28 / 48
Combinatorics on Words Sturmian & Episturmian Words
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}.
Amy Glen (Reykjavík University) Palindromes in Number Theory April 2009 29 / 48
Combinatorics on Words Sturmian & Episturmian Words
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}.
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 .
Amy Glen (Reykjavík University) Palindromes in Number Theory April 2009 29 / 48
Combinatorics on Words Sturmian & Episturmian Words
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}.
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 .
Amy Glen (Reykjavík University) Palindromes in Number Theory April 2009 29 / 48
Combinatorics on Words Sturmian & Episturmian Words
Palindromic closure: Examples
(race)+ =
Amy Glen (Reykjavík University) Palindromes in Number Theory April 2009 30 / 48
Combinatorics on Words Sturmian & Episturmian Words
Palindromic closure: Examples
(race)+ = race
Amy Glen (Reykjavík University) Palindromes in Number Theory April 2009 30 / 48
Combinatorics on Words Sturmian & Episturmian Words
Palindromic closure: Examples
(race)+ = race car
Amy Glen (Reykjavík University) Palindromes in Number Theory April 2009 30 / 48
Combinatorics on Words Sturmian & Episturmian Words
Palindromic closure: Examples
(race)+ = race car
(tie)+ =
Amy Glen (Reykjavík University) Palindromes in Number Theory April 2009 30 / 48
Combinatorics on Words Sturmian & Episturmian Words
Palindromic closure: Examples
(race)+ = race car
(tie)+ = tie
Amy Glen (Reykjavík University) Palindromes in Number Theory April 2009 30 / 48
Combinatorics on Words Sturmian & Episturmian Words
Palindromic closure: Examples
(race)+ = race car
(tie)+ = tie it
Amy Glen (Reykjavík University) Palindromes in Number Theory April 2009 30 / 48
Combinatorics on Words Sturmian & Episturmian Words
Palindromic closure: Examples
(race)+ = race car
(tie)+ = tie it
(tops)+ =
Amy Glen (Reykjavík University) Palindromes in Number Theory April 2009 30 / 48
Combinatorics on Words Sturmian & Episturmian Words
Palindromic closure: Examples
(race)+ = race car
(tie)+ = tie it
(tops)+ = top s
Amy Glen (Reykjavík University) Palindromes in Number Theory April 2009 30 / 48
Combinatorics on Words Sturmian & Episturmian Words
Palindromic closure: Examples
(race)+ = race car
(tie)+ = tie it
(tops)+ = top spot
Amy Glen (Reykjavík University) Palindromes in Number Theory April 2009 30 / 48
Combinatorics on Words Sturmian & Episturmian Words
Palindromic closure: Examples
(race)+ = race car
(tie)+ = tie it
(tops)+ = top spot
(ab)+ =
Amy Glen (Reykjavík University) Palindromes in Number Theory April 2009 30 / 48
Combinatorics on Words Sturmian & Episturmian Words
Palindromic closure: Examples
(race)+ = race car
(tie)+ = tie it
(tops)+ = top spot
(ab)+ = aba
Amy Glen (Reykjavík University) Palindromes in Number Theory April 2009 30 / 48
Combinatorics on Words Sturmian & Episturmian Words
Palindromic closure: Examples
(race)+ = race car
(tie)+ = tie it
(tops)+ = top spot
(ab)+ = aba
Pal(aba) =
Amy Glen (Reykjavík University) Palindromes in Number Theory April 2009 30 / 48
Combinatorics on Words Sturmian & Episturmian Words
Palindromic closure: Examples
(race)+ = race car
(tie)+ = tie it
(tops)+ = top spot
(ab)+ = aba
Pal(aba) = a
Amy Glen (Reykjavík University) Palindromes in Number Theory April 2009 30 / 48
Combinatorics on Words Sturmian & Episturmian Words
Palindromic closure: Examples
(race)+ = race car
(tie)+ = tie it
(tops)+ = top spot
(ab)+ = aba
Pal(aba) = a b
Amy Glen (Reykjavík University) Palindromes in Number Theory April 2009 30 / 48
Combinatorics on Words Sturmian & Episturmian Words
Palindromic closure: Examples
(race)+ = race car
(tie)+ = tie it
(tops)+ = top spot
(ab)+ = aba
Pal(aba) = a b a
Amy Glen (Reykjavík University) Palindromes in Number Theory April 2009 30 / 48
Combinatorics on Words Sturmian & Episturmian Words
Palindromic closure: Examples
(race)+ = race car
(tie)+ = tie it
(tops)+ = top spot
(ab)+ = aba
Pal(aba) = a b a a
Amy Glen (Reykjavík University) Palindromes in Number Theory April 2009 30 / 48
Combinatorics on Words Sturmian & Episturmian Words
Palindromic closure: Examples
(race)+ = race car
(tie)+ = tie it
(tops)+ = top spot
(ab)+ = aba
Pal(aba) = a b a a b a
Amy Glen (Reykjavík University) Palindromes in Number Theory April 2009 30 / 48
Combinatorics on Words Sturmian & Episturmian Words
Palindromic closure: Examples
(race)+ = race car
(tie)+ = tie it
(tops)+ = top spot
(ab)+ = aba
Pal(aba) = a b a a b a
Pal(abc) = a b a c a b a
Amy Glen (Reykjavík University) Palindromes in Number Theory April 2009 30 / 48
Combinatorics on Words Sturmian & Episturmian Words
Palindromic closure: Examples
(race)+ = race car
(tie)+ = tie it
(tops)+ = top spot
(ab)+ = aba
Pal(aba) = a b a a b a
Pal(abc) = a b a c a b a
Pal(race) =
Amy Glen (Reykjavík University) Palindromes in Number Theory April 2009 30 / 48
Combinatorics on Words Sturmian & Episturmian Words
Palindromic closure: Examples
(race)+ = race car
(tie)+ = tie it
(tops)+ = top spot
(ab)+ = aba
Pal(aba) = a b a a b a
Pal(abc) = a b a c a b a
Pal(race) = rarcrarerarcrar
Amy Glen (Reykjavík University) Palindromes in Number Theory April 2009 30 / 48
Combinatorics on Words Sturmian & Episturmian Words
Palindromic closure: Examples
(race)+ = race car
(tie)+ = tie it
(tops)+ = top spot
(ab)+ = aba
Pal(aba) = a b a a b a
Pal(abc) = a b a c a b a
Pal(race) = rarcrarerarcrar
L(5, 3) = aabaabab = aPal(aba)b
Amy Glen (Reykjavík University) Palindromes in Number Theory April 2009 30 / 48
Combinatorics on Words Sturmian & Episturmian Words
Palindromic closure: Examples
(race)+ = race car
(tie)+ = tie it
(tops)+ = top spot
(ab)+ = aba
Pal(aba) = a b a a b a
Pal(abc) = a b a c a b a
Pal(race) = rarcrarerarcrar
L(5, 3) = aabaabab = aPal(aba)b
L(7, 4) = aabaabaabab = aPal(abaa)b
Amy Glen (Reykjavík University) Palindromes in Number Theory April 2009 30 / 48
Combinatorics on Words Sturmian & Episturmian Words
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(∆).
Amy Glen (Reykjavík University) Palindromes in Number Theory April 2009 31 / 48
Combinatorics on Words Sturmian & Episturmian Words
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(∆).
∆: directive word of s.
Amy Glen (Reykjavík University) Palindromes in Number Theory April 2009 31 / 48
Combinatorics on Words Sturmian & Episturmian Words
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(∆).
∆: directive word of s.
Example: Fibonacci word is directed by ∆ = (ab)(ab)(ab) · · · .
Amy Glen (Reykjavík University) Palindromes in Number Theory April 2009 31 / 48
Combinatorics on Words Sturmian & Episturmian Words
Recall: Fibonacci word
a a
a a
a
a a
b
b
b
b
Line of slope√
5−12 −→ Fibonacci word
Amy Glen (Reykjavík University) Palindromes in Number Theory April 2009 32 / 48
Combinatorics on Words Sturmian & Episturmian Words
Recall: Fibonacci word
a a
a a
a
a a
b
b
b
b
Line of slope√
5−12 −→ Fibonacci word
∆ = (ab)(ab)(ab) · · · −→ f = a
Amy Glen (Reykjavík University) Palindromes in Number Theory April 2009 32 / 48
Combinatorics on Words Sturmian & Episturmian Words
Recall: Fibonacci word
a a
a a
a
a a
b
b
b
b
Line of slope√
5−12 −→ Fibonacci word
∆ = (ab)(ab)(ab) · · · −→ f = ab
Amy Glen (Reykjavík University) Palindromes in Number Theory April 2009 32 / 48
Combinatorics on Words Sturmian & Episturmian Words
Recall: Fibonacci word
a a
a a
a
a a
b
b
b
b
Line of slope√
5−12 −→ Fibonacci word
∆ = (ab)(ab)(ab) · · · −→ f = aba
Amy Glen (Reykjavík University) Palindromes in Number Theory April 2009 32 / 48
Combinatorics on Words Sturmian & Episturmian Words
Recall: Fibonacci word
a a
a a
a
a a
b
b
b
b
Line of slope√
5−12 −→ Fibonacci word
∆ = (ab)(ab)(ab) · · · −→ f = abaa
Amy Glen (Reykjavík University) Palindromes in Number Theory April 2009 32 / 48
Combinatorics on Words Sturmian & Episturmian Words
Recall: Fibonacci word
a a
a a
a
a a
b
b
b
b
Line of slope√
5−12 −→ Fibonacci word
∆ = (ab)(ab)(ab) · · · −→ f = abaaba
Amy Glen (Reykjavík University) Palindromes in Number Theory April 2009 32 / 48
Combinatorics on Words Sturmian & Episturmian Words
Recall: Fibonacci word
a a
a a
a
a a
b
b
b
b
Line of slope√
5−12 −→ Fibonacci word
∆ = (ab)(ab)(ab) · · · −→ f = abaabab
Amy Glen (Reykjavík University) Palindromes in Number Theory April 2009 32 / 48
Combinatorics on Words Sturmian & Episturmian Words
Recall: Fibonacci word
a a
a a
a
a a
b
b
b
b
Line of slope√
5−12 −→ Fibonacci word
∆ = (ab)(ab)(ab) · · · −→ f = abaababaaba · · ·
Amy Glen (Reykjavík University) Palindromes in Number Theory April 2009 32 / 48
Combinatorics on Words Sturmian & Episturmian Words
Recall: Fibonacci word
a a
a a
a
a a
b
b
b
b
Line of slope√
5−12 −→ Fibonacci word
∆ = (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.
Amy Glen (Reykjavík University) Palindromes in Number Theory April 2009 32 / 48
Combinatorics on Words Sturmian & Episturmian Words
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(∆).
Amy Glen (Reykjavík University) Palindromes in Number Theory April 2009 33 / 48
Combinatorics on Words Sturmian & Episturmian Words
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(∆).
Example: ∆ = (abc)(abc)(abc) · · · directs the Tribonacci word:
Amy Glen (Reykjavík University) Palindromes in Number Theory April 2009 33 / 48
Combinatorics on Words Sturmian & Episturmian Words
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(∆).
Example: ∆ = (abc)(abc)(abc) · · · directs the Tribonacci word:
r = a
Amy Glen (Reykjavík University) Palindromes in Number Theory April 2009 33 / 48
Combinatorics on Words Sturmian & Episturmian Words
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(∆).
Example: ∆ = (abc)(abc)(abc) · · · directs the Tribonacci word:
r = ab
Amy Glen (Reykjavík University) Palindromes in Number Theory April 2009 33 / 48
Combinatorics on Words Sturmian & Episturmian Words
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(∆).
Example: ∆ = (abc)(abc)(abc) · · · directs the Tribonacci word:
r = aba
Amy Glen (Reykjavík University) Palindromes in Number Theory April 2009 33 / 48
Combinatorics on Words Sturmian & Episturmian Words
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(∆).
Example: ∆ = (abc)(abc)(abc) · · · directs the Tribonacci word:
r = abac
Amy Glen (Reykjavík University) Palindromes in Number Theory April 2009 33 / 48
Combinatorics on Words Sturmian & Episturmian Words
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(∆).
Example: ∆ = (abc)(abc)(abc) · · · directs the Tribonacci word:
r = abacaba
Amy Glen (Reykjavík University) Palindromes in Number Theory April 2009 33 / 48
Combinatorics on Words Sturmian & Episturmian Words
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(∆).
Example: ∆ = (abc)(abc)(abc) · · · directs the Tribonacci word:
r = abacabaa
Amy Glen (Reykjavík University) Palindromes in Number Theory April 2009 33 / 48
Combinatorics on Words Sturmian & Episturmian Words
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(∆).
Example: ∆ = (abc)(abc)(abc) · · · directs the Tribonacci word:
r = abacabaabacaba
Amy Glen (Reykjavík University) Palindromes in Number Theory April 2009 33 / 48
Combinatorics on Words Sturmian & Episturmian Words
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(∆).
Example: ∆ = (abc)(abc)(abc) · · · directs the Tribonacci word:
r = abacabaabacabab
Amy Glen (Reykjavík University) Palindromes in Number Theory April 2009 33 / 48
Combinatorics on Words Sturmian & Episturmian Words
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(∆).
Example: ∆ = (abc)(abc)(abc) · · · directs the Tribonacci word:
r = abacabaabacababacabaabacaba
Amy Glen (Reykjavík University) Palindromes in Number Theory April 2009 33 / 48
Combinatorics on Words Sturmian & Episturmian Words
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(∆).
Example: ∆ = (abc)(abc)(abc) · · · directs the Tribonacci word:
r = abacabaabacababacabaabacabac
Amy Glen (Reykjavík University) Palindromes in Number Theory April 2009 33 / 48
Combinatorics on Words Sturmian & Episturmian Words
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(∆).
Example: ∆ = (abc)(abc)(abc) · · · directs the Tribonacci word:
r = abacabaabacababacabaabacabacabaabaca · · ·
Amy Glen (Reykjavík University) Palindromes in Number Theory April 2009 33 / 48
Combinatorics on Words Sturmian & Episturmian Words
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(∆).
Example: ∆ = (abc)(abc)(abc) · · · directs the Tribonacci word:
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.
Amy Glen (Reykjavík University) Palindromes in Number Theory April 2009 33 / 48
Some Connections to Number Theory
Outline
1 Combinatorics on WordsSturmian & Episturmian Words
2 Some Connections to Number TheoryContinued Fractions & Sturmian WordsPalindromes & Diophantine ApproximationTranscendental NumbersMiscellaneous
Amy Glen (Reykjavík University) Palindromes in Number Theory April 2009 34 / 48
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.
Amy Glen (Reykjavík University) Palindromes in Number Theory April 2009 35 / 48
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.
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, . . .
Amy Glen (Reykjavík University) Palindromes in Number Theory April 2009 35 / 48
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.
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.
Amy Glen (Reykjavík University) Palindromes in Number Theory April 2009 35 / 48
Some Connections to Number Theory Continued Fractions & Sturmian Words
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.
Amy Glen (Reykjavík University) Palindromes in Number Theory April 2009 36 / 48
Some Connections to Number Theory Continued Fractions & Sturmian Words
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.
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
Amy Glen (Reykjavík University) Palindromes in Number Theory April 2009 36 / 48
Some Connections to Number Theory Continued Fractions & Sturmian Words
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.
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
f = ab s1, length F2 = 2
Amy Glen (Reykjavík University) Palindromes in Number Theory April 2009 36 / 48
Some Connections to Number Theory Continued Fractions & Sturmian Words
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.
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
f = aba s2, length F3 = 3
Amy Glen (Reykjavík University) Palindromes in Number Theory April 2009 36 / 48
Some Connections to Number Theory Continued Fractions & Sturmian Words
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.
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
f = abaab s3, length F4 = 5
Amy Glen (Reykjavík University) Palindromes in Number Theory April 2009 36 / 48
Some Connections to Number Theory Continued Fractions & Sturmian Words
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.
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
f = abaababa s4, length F5 = 8
Amy Glen (Reykjavík University) Palindromes in Number Theory April 2009 36 / 48
Some Connections to Number Theory Continued Fractions & Sturmian Words
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.
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
f = abaababaabaab s5, length F6 = 13
Amy Glen (Reykjavík University) Palindromes in Number Theory April 2009 36 / 48
Some Connections to Number Theory Continued Fractions & Sturmian Words
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.
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
f = abaababaabaababaababa s6, length F7 = 21
Amy Glen (Reykjavík University) Palindromes in Number Theory April 2009 36 / 48
Some Connections to Number Theory Continued Fractions & Sturmian Words
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.
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
f = abaababaabaababaababa · · ·
Amy Glen (Reykjavík University) Palindromes in Number Theory April 2009 36 / 48
Some Connections to Number Theory Continued Fractions & Sturmian Words
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.
In general:
|sn| = qn+1 for all n
Amy Glen (Reykjavík University) Palindromes in Number Theory April 2009 36 / 48
Some Connections to Number Theory Continued Fractions & Sturmian Words
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.
In general:
|sn| = qn+1 for all n
sn = Pal(v)xy for some v ∈ {a, b}∗ and {x , y} = {a, b}
Amy Glen (Reykjavík University) Palindromes in Number Theory April 2009 36 / 48
Some Connections to Number Theory Continued Fractions & Sturmian Words
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.
In general:
|sn| = qn+1 for all n
sn = Pal(v)xy for some v ∈ {a, b}∗ and {x , y} = {a, b}cα = Pal(ad1bd2ad3bd4 · · · ) E.g. f = Pal(abababa . . .)
Amy Glen (Reykjavík University) Palindromes in Number Theory April 2009 36 / 48
Some Connections to Number Theory Continued Fractions & Sturmian Words
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.
In general:
|sn| = qn+1 for all n
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.
Amy Glen (Reykjavík University) Palindromes in Number Theory April 2009 36 / 48
Some Connections to Number Theory Palindromes & Diophantine Approximation
Basics of Diophantine Approximation
Diophantine Approximation is concerned with the approximation ofreal numbers by rational numbers.
Amy Glen (Reykjavík University) Palindromes in Number Theory April 2009 37 / 48
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?
Amy Glen (Reykjavík University) Palindromes in Number Theory April 2009 37 / 48
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.
Amy Glen (Reykjavík University) Palindromes in Number Theory April 2009 37 / 48
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)).
Amy Glen (Reykjavík University) Palindromes in Number Theory April 2009 37 / 48
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.
Amy Glen (Reykjavík University) Palindromes in Number Theory April 2009 37 / 48
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.
Amy Glen (Reykjavík University) Palindromes in Number Theory April 2009 37 / 48
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.
CF theory → for every irrational number ξ, the inequality|ξ − p/q| < 1/q2 always has infinitely many rational solutions p/q.
Amy Glen (Reykjavík University) Palindromes in Number Theory April 2009 37 / 48
Some Connections to Number Theory Palindromes & Diophantine Approximation
Palindromes & Diophantine Approximation
Roy 2003, 2004: Introduced a “palindromic prefix method” in thestudy of simultaneous approximation to a real number and its square.
Amy Glen (Reykjavík University) Palindromes in Number Theory April 2009 38 / 48
Some Connections to Number Theory Palindromes & Diophantine Approximation
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”.
Amy Glen (Reykjavík University) Palindromes in Number Theory April 2009 38 / 48
Some Connections to Number Theory Palindromes & Diophantine Approximation
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]
Amy Glen (Reykjavík University) Palindromes in Number Theory April 2009 38 / 48
Some Connections to Number Theory Palindromes & Diophantine Approximation
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.
Amy Glen (Reykjavík University) Palindromes in Number Theory April 2009 38 / 48
Some Connections to Number Theory Palindromes & Diophantine Approximation
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.
Fischler introduced palindromic prefix density . . .
Amy Glen (Reykjavík University) Palindromes in Number Theory April 2009 38 / 48
Some Connections to Number Theory Palindromes & Diophantine Approximation
Palindromic prefix density
Let w = w1w2w3 · · · be an infinite word (with each wi a letter).
Amy Glen (Reykjavík University) Palindromes in Number Theory April 2009 39 / 48
Some Connections to Number Theory Palindromes & Diophantine Approximation
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.
Amy Glen (Reykjavík University) Palindromes in Number Theory April 2009 39 / 48
Some Connections to Number Theory Palindromes & Diophantine Approximation
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.
Amy Glen (Reykjavík University) Palindromes in Number Theory April 2009 39 / 48
Some Connections to Number Theory Palindromes & Diophantine Approximation
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.
Note: 0 ≤ dp(w) ≤ 1 for any infinite word w.
Amy Glen (Reykjavík University) Palindromes in Number Theory April 2009 39 / 48
Some Connections to Number Theory Palindromes & Diophantine Approximation
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.
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.
Amy Glen (Reykjavík University) Palindromes in Number Theory April 2009 39 / 48
Some Connections to Number Theory Palindromes & Diophantine Approximation
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).
Amy Glen (Reykjavík University) Palindromes in Number Theory April 2009 40 / 48
Some Connections to Number Theory Palindromes & Diophantine Approximation
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).
Fischler’s bound is optimal . . .
Amy Glen (Reykjavík University) Palindromes in Number Theory April 2009 40 / 48
Some Connections to Number Theory Palindromes & Diophantine Approximation
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).
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.
Amy Glen (Reykjavík University) Palindromes in Number Theory April 2009 40 / 48
Some Connections to Number Theory Palindromes & Diophantine Approximation
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 . . .
Amy Glen (Reykjavík University) Palindromes in Number Theory April 2009 41 / 48
Some Connections to Number Theory Palindromes & Diophantine Approximation
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 . . .
Amy Glen (Reykjavík University) Palindromes in Number Theory April 2009 41 / 48
Some Connections to Number Theory Palindromes & Diophantine Approximation
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 . . .
Fischler 2006: Introduced a natural generalisation of standardepisturmian words . . .
Amy Glen (Reykjavík University) Palindromes in Number Theory April 2009 41 / 48
Some Connections to Number Theory Palindromes & Diophantine Approximation
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.
Amy Glen (Reykjavík University) Palindromes in Number Theory April 2009 42 / 48
Some Connections to Number Theory Palindromes & Diophantine Approximation
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 .
Amy Glen (Reykjavík University) Palindromes in Number Theory April 2009 42 / 48
Some Connections to Number Theory Palindromes & Diophantine Approximation
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.
Amy Glen (Reykjavík University) Palindromes in Number Theory April 2009 42 / 48
Some Connections to Number Theory Palindromes & Diophantine Approximation
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.
Amy Glen (Reykjavík University) Palindromes in Number Theory April 2009 42 / 48
Some Connections to Number Theory Palindromes & Diophantine Approximation
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.
Example: (abcacba)∞ = abcacbaabcacbaabcacba · · · has abundantpalindromic prefixes, but it is not standard episturmian.
Amy Glen (Reykjavík University) Palindromes in Number Theory April 2009 42 / 48
Some Connections to Number Theory Palindromes & Diophantine Approximation
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.
Amy Glen (Reykjavík University) Palindromes in Number Theory April 2009 43 / 48
Some Connections to Number Theory Palindromes & Diophantine Approximation
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.
Amy Glen (Reykjavík University) Palindromes in Number Theory April 2009 43 / 48
Some Connections to Number Theory Palindromes & Diophantine Approximation
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 · · ·
Amy Glen (Reykjavík University) Palindromes in Number Theory April 2009 43 / 48
Some Connections to Number Theory Palindromes & Diophantine Approximation
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 · · ·abbbbbb · · ·
Amy Glen (Reykjavík University) Palindromes in Number Theory April 2009 43 / 48
Some Connections to Number Theory Palindromes & Diophantine Approximation
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 · · ·abbbbbb · · ·abaabaaabaaaab · · ·
Amy Glen (Reykjavík University) Palindromes in Number Theory April 2009 43 / 48
Some Connections to Number Theory Palindromes & Diophantine Approximation
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 · · ·abbbbbb · · ·abaabaaabaaaab · · ·(abcba)(abcba)(abcba) · · ·
Amy Glen (Reykjavík University) Palindromes in Number Theory April 2009 43 / 48
Some Connections to Number Theory Palindromes & Diophantine Approximation
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 · · ·abbbbbb · · ·abaabaaabaaaab · · ·(abcba)(abcba)(abcba) · · ·Sturmian and episturmian words
Amy Glen (Reykjavík University) Palindromes in Number Theory April 2009 43 / 48
Some Connections to Number Theory Palindromes & Diophantine Approximation
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 · · ·abbbbbb · · ·abaabaaabaaaab · · ·(abcba)(abcba)(abcba) · · ·Sturmian and episturmian words
Infinite words with abundant palindromic prefixes
Amy Glen (Reykjavík University) Palindromes in Number Theory April 2009 43 / 48
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.
Amy Glen (Reykjavík University) Palindromes in Number Theory April 2009 44 / 48
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.
Amy Glen (Reykjavík University) Palindromes in Number Theory April 2009 44 / 48
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)
Amy Glen (Reykjavík University) Palindromes in Number Theory April 2009 44 / 48
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)
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
>
>
>
>
>
>
>
=
>
>
>
>
>
>
>
;
transcendence criteria from DA
Amy Glen (Reykjavík University) Palindromes in Number Theory April 2009 44 / 48
Some Connections to Number Theory Transcendental Numbers
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.
Amy Glen (Reykjavík University) Palindromes in Number Theory April 2009 45 / 48
Some Connections to Number Theory Transcendental Numbers
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.
Amy Glen (Reykjavík University) Palindromes in Number Theory April 2009 45 / 48
Some Connections to Number Theory Transcendental Numbers
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.
Amy Glen (Reykjavík University) Palindromes in Number Theory April 2009 45 / 48
Some Connections to Number Theory Transcendental Numbers
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.
Proved by showing that Sturmian continued fractions admit very goodapproximations by quadratic real numbers.
Amy Glen (Reykjavík University) Palindromes in Number Theory April 2009 45 / 48
Some Connections to Number Theory Transcendental Numbers
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.
Proved by showing that Sturmian continued fractions admit very goodapproximations by quadratic real numbers.
In particular, they used the fact that any Sturmian word begins witharbitrarily long squares (words of the form XX = X 2).
Amy Glen (Reykjavík University) Palindromes in Number Theory April 2009 45 / 48
Some Connections to Number Theory Transcendental Numbers
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.
Proved by showing that Sturmian continued fractions admit very goodapproximations by quadratic real numbers.
In particular, they used the fact that any Sturmian word begins witharbitrarily long squares (words of the form XX = X 2).
Example: f = aba · ababaabaababaababaabaababaaba · · ·Amy Glen (Reykjavík University) Palindromes in Number Theory April 2009 45 / 48
Some Connections to Number Theory Transcendental Numbers
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.
Proved by showing that Sturmian continued fractions admit very goodapproximations by quadratic real numbers.
In particular, they used the fact that any Sturmian word begins witharbitrarily long squares (words of the form XX = X 2).
Example: f = abaab · abaabaababaababaabaababaaba · · ·Amy Glen (Reykjavík University) Palindromes in Number Theory April 2009 45 / 48
Some Connections to Number Theory Transcendental Numbers
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.
Proved by showing that Sturmian continued fractions admit very goodapproximations by quadratic real numbers.
In particular, they used the fact that any Sturmian word begins witharbitrarily long squares (words of the form XX = X 2).
Example: f = abaababa · abaababaababaabaababaaba · · ·Amy Glen (Reykjavík University) Palindromes in Number Theory April 2009 45 / 48
Some Connections to Number Theory Transcendental Numbers
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.
Proved by showing that Sturmian continued fractions admit very goodapproximations by quadratic real numbers.
In particular, they used the fact that any Sturmian word begins witharbitrarily long squares (words of the form XX = X 2).
Example: f = abaababaabaab · abaababaabaababaaba · · ·Amy Glen (Reykjavík University) Palindromes in Number Theory April 2009 45 / 48
Some Connections to Number Theory Transcendental Numbers
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 . . .
Amy Glen (Reykjavík University) Palindromes in Number Theory April 2009 46 / 48
Some Connections to Number Theory Transcendental Numbers
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.
Amy Glen (Reykjavík University) Palindromes in Number Theory April 2009 46 / 48
Some Connections to Number Theory Transcendental Numbers
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.
Amy Glen (Reykjavík University) Palindromes in Number Theory April 2009 46 / 48
Some Connections to Number Theory Transcendental Numbers
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.
Proof of theorem rests on Schmidt’s Subspace Theorem (1972).
Amy Glen (Reykjavík University) Palindromes in Number Theory April 2009 46 / 48
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.
Amy Glen (Reykjavík University) Palindromes in Number Theory April 2009 47 / 48
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.
Strict episturmian words (or Arnoux-Rauzy sequences) naturallygeneralise and extend this rich interplay. [Rauzy 1982, Arnoux-Rauzy 1991]
Amy Glen (Reykjavík University) Palindromes in Number Theory April 2009 47 / 48
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.
Strict episturmian words (or Arnoux-Rauzy sequences) naturallygeneralise and extend this rich interplay. [Rauzy 1982, Arnoux-Rauzy 1991]
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 · · · ).
Amy Glen (Reykjavík University) Palindromes in Number Theory April 2009 47 / 48
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.
Strict episturmian words (or Arnoux-Rauzy sequences) naturallygeneralise and extend this rich interplay. [Rauzy 1982, Arnoux-Rauzy 1991]
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 · · · ).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]
Amy Glen (Reykjavík University) Palindromes in Number Theory April 2009 47 / 48
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.
Strict episturmian words (or Arnoux-Rauzy sequences) naturallygeneralise and extend this rich interplay. [Rauzy 1982, Arnoux-Rauzy 1991]
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 · · · ).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.
Amy Glen (Reykjavík University) Palindromes in Number Theory April 2009 47 / 48
Some Connections to Number Theory Miscellaneous
Thank you!
Amy Glen (Reykjavík University) Palindromes in Number Theory April 2009 48 / 48