combinatorics on words & some applications in number theory
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Combinatorics on Words
&
Some Applications in Number Theory
Amy Glen
School of Chemical & Mathematical Sciences
Murdoch University, Perth, Australia
amy.glen@gmail.comhttp://wwwstaff.murdoch.edu.au/∼aglen
Groups & Combinatorics Seminar series @ UWA
Amy Glen (Murdoch University) CoW & Number Theory November 17, 2009 1 / 51
Outline
1 Combinatorics on WordsSturmian & Episturmian Words
2 Some Applications in Number TheoryContinued Fractions & Sturmian WordsDistribution modulo 1 & Sturmian WordsTranscendental Numbers
Amy Glen (Murdoch University) CoW & Number Theory November 17, 2009 2 / 51
Combinatorics on Words
Outline
1 Combinatorics on WordsSturmian & Episturmian Words
2 Some Applications in Number TheoryContinued Fractions & Sturmian WordsDistribution modulo 1 & Sturmian WordsTranscendental Numbers
Amy Glen (Murdoch University) CoW & Number Theory November 17, 2009 3 / 51
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 (Murdoch University) CoW & Number Theory November 17, 2009 4 / 51
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 (Murdoch University) CoW & Number Theory November 17, 2009 4 / 51
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 (Murdoch University) CoW & Number Theory November 17, 2009 4 / 51
Combinatorics on Words
Combinatorics on words: A brief history
Relatively new area of Discrete Mathematics
Amy Glen (Murdoch University) CoW & Number Theory November 17, 2009 5 / 51
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 (Murdoch University) CoW & Number Theory November 17, 2009 5 / 51
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 (Murdoch University) CoW & Number Theory November 17, 2009 5 / 51
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 (Murdoch University) CoW & Number Theory November 17, 2009 5 / 51
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 (Murdoch University) CoW & Number Theory November 17, 2009 5 / 51
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 (Murdoch University) CoW & Number Theory November 17, 2009 6 / 51
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 (Adelaide suburb).
Amy Glen (Murdoch University) CoW & Number Theory November 17, 2009 6 / 51
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 (Adelaide suburb).
The extent to which a word exhibits strong regularity properties isgenerally inversely proportional to its “complexity”.
Amy Glen (Murdoch University) CoW & Number Theory November 17, 2009 6 / 51
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 (Adelaide suburb).
The extent to which a word exhibits strong regularity properties isgenerally inversely proportional to its “complexity”.
Basic measure: Number of distinct blocks (factors) of each lengthoccurring in the word.
Amy Glen (Murdoch University) CoW & Number Theory November 17, 2009 6 / 51
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 (Adelaide suburb).
The extent to which a word exhibits strong regularity properties isgenerally inversely proportional to its “complexity”.
Basic measure: Number of distinct blocks (factors) of each lengthoccurring in the word.
Example: w = abca has 9 distinct factors:
a, b, c ,︸ ︷︷ ︸
1
ab, bc , ca,︸ ︷︷ ︸
2
abc , bca,︸ ︷︷ ︸
3
abca.︸ ︷︷ ︸
4
Amy Glen (Murdoch University) CoW & Number Theory November 17, 2009 6 / 51
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 (Murdoch University) CoW & Number Theory November 17, 2009 7 / 51
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 (Murdoch University) CoW & Number Theory November 17, 2009 7 / 51
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 (Murdoch University) CoW & Number Theory November 17, 2009 7 / 51
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 (Murdoch University) CoW & Number Theory November 17, 2009 7 / 51
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 (Murdoch University) CoW & Number Theory November 17, 2009 7 / 51
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 (Murdoch University) CoW & Number Theory November 17, 2009 7 / 51
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 (Murdoch University) CoW & Number Theory November 17, 2009 7 / 51
Combinatorics on Words Sturmian & Episturmian Words
Christoffel words: A special family of finite Sturmian words
Lower Christoffel word of slope 35
Amy Glen (Murdoch University) CoW & Number Theory November 17, 2009 8 / 51
Combinatorics on Words Sturmian & Episturmian Words
Christoffel words: Construction
Lower Christoffel word of slope 35
Amy Glen (Murdoch University) CoW & Number Theory November 17, 2009 9 / 51
Combinatorics on Words Sturmian & Episturmian Words
Christoffel words: Construction
Lower Christoffel word of slope 35
Amy Glen (Murdoch University) CoW & Number Theory November 17, 2009 10 / 51
Combinatorics on Words Sturmian & Episturmian Words
Christoffel words: Construction
Lower Christoffel word of slope 35
Amy Glen (Murdoch University) CoW & Number Theory November 17, 2009 11 / 51
Combinatorics on Words Sturmian & Episturmian Words
Christoffel words: Construction
Lower Christoffel word of slope 35
Amy Glen (Murdoch University) CoW & Number Theory November 17, 2009 12 / 51
Combinatorics on Words Sturmian & Episturmian Words
Christoffel words: Construction
Lower Christoffel word of slope 35
Amy Glen (Murdoch University) CoW & Number Theory November 17, 2009 13 / 51
Combinatorics on Words Sturmian & Episturmian Words
Christoffel words: Construction
Lower Christoffel word of slope 35
Amy Glen (Murdoch University) CoW & Number Theory November 17, 2009 14 / 51
Combinatorics on Words Sturmian & Episturmian Words
Christoffel words: Construction
Lower Christoffel word of slope 35
Amy Glen (Murdoch University) CoW & Number Theory November 17, 2009 15 / 51
Combinatorics on Words Sturmian & Episturmian Words
Christoffel words: Construction
Lower Christoffel word of slope 35
Amy Glen (Murdoch University) CoW & Number Theory November 17, 2009 16 / 51
Combinatorics on Words Sturmian & Episturmian Words
Christoffel words: Construction
Lower Christoffel word of slope 35
a
L(5,3) = a
Amy Glen (Murdoch University) CoW & Number Theory November 17, 2009 17 / 51
Combinatorics on Words Sturmian & Episturmian Words
Christoffel words: Construction
Lower Christoffel word of slope 35
a a
L(5,3) = aa
Amy Glen (Murdoch University) CoW & Number Theory November 17, 2009 18 / 51
Combinatorics on Words Sturmian & Episturmian Words
Christoffel words: Construction
Lower Christoffel word of slope 35
a a
b
L(5,3) = aab
Amy Glen (Murdoch University) CoW & Number Theory November 17, 2009 19 / 51
Combinatorics on Words Sturmian & Episturmian Words
Christoffel words: Construction
Lower Christoffel word of slope 35
a a
ba
L(5,3) = aaba
Amy Glen (Murdoch University) CoW & Number Theory November 17, 2009 20 / 51
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 (Murdoch University) CoW & Number Theory November 17, 2009 21 / 51
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 (Murdoch University) CoW & Number Theory November 17, 2009 22 / 51
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 (Murdoch University) CoW & Number Theory November 17, 2009 23 / 51
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 (Murdoch University) CoW & Number Theory November 17, 2009 24 / 51
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 (Murdoch University) CoW & Number Theory November 17, 2009 25 / 51
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 (Murdoch University) CoW & Number Theory November 17, 2009 26 / 51
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 (Murdoch University) CoW & Number Theory November 17, 2009 26 / 51
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 (Murdoch University) CoW & Number Theory November 17, 2009 27 / 51
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 (Murdoch University) CoW & Number Theory November 17, 2009 27 / 51
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 (Murdoch University) CoW & Number Theory November 17, 2009 28 / 51
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 (Murdoch University) CoW & Number Theory November 17, 2009 28 / 51
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 (Murdoch University) CoW & Number Theory November 17, 2009 28 / 51
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 (Murdoch University) CoW & Number Theory November 17, 2009 28 / 51
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 (Murdoch University) CoW & Number Theory November 17, 2009 28 / 51
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 = bpawhere p = Pal(v) for some word v over {a, b}.
Amy Glen (Murdoch University) CoW & Number Theory November 17, 2009 29 / 51
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 = bpawhere 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 (Murdoch University) CoW & Number Theory November 17, 2009 29 / 51
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 = bpawhere 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 (Murdoch University) CoW & Number Theory November 17, 2009 29 / 51
Combinatorics on Words Sturmian & Episturmian Words
Palindromic closure: Examples
(race)+ =
Amy Glen (Murdoch University) CoW & Number Theory November 17, 2009 30 / 51
Combinatorics on Words Sturmian & Episturmian Words
Palindromic closure: Examples
(race)+ = race
Amy Glen (Murdoch University) CoW & Number Theory November 17, 2009 30 / 51
Combinatorics on Words Sturmian & Episturmian Words
Palindromic closure: Examples
(race)+ = race car
Amy Glen (Murdoch University) CoW & Number Theory November 17, 2009 30 / 51
Combinatorics on Words Sturmian & Episturmian Words
Palindromic closure: Examples
(race)+ = race car
(tie)+ =
Amy Glen (Murdoch University) CoW & Number Theory November 17, 2009 30 / 51
Combinatorics on Words Sturmian & Episturmian Words
Palindromic closure: Examples
(race)+ = race car
(tie)+ = tie
Amy Glen (Murdoch University) CoW & Number Theory November 17, 2009 30 / 51
Combinatorics on Words Sturmian & Episturmian Words
Palindromic closure: Examples
(race)+ = race car
(tie)+ = tie it
Amy Glen (Murdoch University) CoW & Number Theory November 17, 2009 30 / 51
Combinatorics on Words Sturmian & Episturmian Words
Palindromic closure: Examples
(race)+ = race car
(tie)+ = tie it
(tops)+ =
Amy Glen (Murdoch University) CoW & Number Theory November 17, 2009 30 / 51
Combinatorics on Words Sturmian & Episturmian Words
Palindromic closure: Examples
(race)+ = race car
(tie)+ = tie it
(tops)+ = top s
Amy Glen (Murdoch University) CoW & Number Theory November 17, 2009 30 / 51
Combinatorics on Words Sturmian & Episturmian Words
Palindromic closure: Examples
(race)+ = race car
(tie)+ = tie it
(tops)+ = top spot
Amy Glen (Murdoch University) CoW & Number Theory November 17, 2009 30 / 51
Combinatorics on Words Sturmian & Episturmian Words
Palindromic closure: Examples
(race)+ = race car
(tie)+ = tie it
(tops)+ = top spot
Pal(aba) =
Amy Glen (Murdoch University) CoW & Number Theory November 17, 2009 30 / 51
Combinatorics on Words Sturmian & Episturmian Words
Palindromic closure: Examples
(race)+ = race car
(tie)+ = tie it
(tops)+ = top spot
Pal(aba) = a
Amy Glen (Murdoch University) CoW & Number Theory November 17, 2009 30 / 51
Combinatorics on Words Sturmian & Episturmian Words
Palindromic closure: Examples
(race)+ = race car
(tie)+ = tie it
(tops)+ = top spot
Pal(aba) = a b
Amy Glen (Murdoch University) CoW & Number Theory November 17, 2009 30 / 51
Combinatorics on Words Sturmian & Episturmian Words
Palindromic closure: Examples
(race)+ = race car
(tie)+ = tie it
(tops)+ = top spot
Pal(aba) = a b a
Amy Glen (Murdoch University) CoW & Number Theory November 17, 2009 30 / 51
Combinatorics on Words Sturmian & Episturmian Words
Palindromic closure: Examples
(race)+ = race car
(tie)+ = tie it
(tops)+ = top spot
Pal(aba) = a b a a
Amy Glen (Murdoch University) CoW & Number Theory November 17, 2009 30 / 51
Combinatorics on Words Sturmian & Episturmian Words
Palindromic closure: Examples
(race)+ = race car
(tie)+ = tie it
(tops)+ = top spot
Pal(aba) = a b a a b a
Amy Glen (Murdoch University) CoW & Number Theory November 17, 2009 30 / 51
Combinatorics on Words Sturmian & Episturmian Words
Palindromic closure: Examples
(race)+ = race car
(tie)+ = tie it
(tops)+ = top spot
Pal(aba) = a b a a b a
Pal(abc) = a b a c a b a
Amy Glen (Murdoch University) CoW & Number Theory November 17, 2009 30 / 51
Combinatorics on Words Sturmian & Episturmian Words
Palindromic closure: Examples
(race)+ = race car
(tie)+ = tie it
(tops)+ = top spot
Pal(aba) = a b a a b a
Pal(abc) = a b a c a b a
Pal(race) =rarcrarerarcrar
Amy Glen (Murdoch University) CoW & Number Theory November 17, 2009 30 / 51
Combinatorics on Words Sturmian & Episturmian Words
Palindromic closure: Examples
(race)+ = race car
(tie)+ = tie it
(tops)+ = top spot
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 (Murdoch University) CoW & Number Theory November 17, 2009 30 / 51
Combinatorics on Words Sturmian & Episturmian Words
Palindromic closure: Examples
(race)+ = race car
(tie)+ = tie it
(tops)+ = top spot
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 (Murdoch University) CoW & Number Theory November 17, 2009 30 / 51
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 (Murdoch University) CoW & Number Theory November 17, 2009 31 / 51
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 (Murdoch University) CoW & Number Theory November 17, 2009 31 / 51
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 (Murdoch University) CoW & Number Theory November 17, 2009 31 / 51
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 (Murdoch University) CoW & Number Theory November 17, 2009 32 / 51
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 (Murdoch University) CoW & Number Theory November 17, 2009 32 / 51
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 (Murdoch University) CoW & Number Theory November 17, 2009 32 / 51
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 (Murdoch University) CoW & Number Theory November 17, 2009 32 / 51
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 (Murdoch University) CoW & Number Theory November 17, 2009 32 / 51
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 (Murdoch University) CoW & Number Theory November 17, 2009 32 / 51
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 (Murdoch University) CoW & Number Theory November 17, 2009 32 / 51
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 (Murdoch University) CoW & Number Theory November 17, 2009 32 / 51
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 (Murdoch University) CoW & Number Theory November 17, 2009 32 / 51
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 (Murdoch University) CoW & Number Theory November 17, 2009 33 / 51
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 (Murdoch University) CoW & Number Theory November 17, 2009 33 / 51
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 (Murdoch University) CoW & Number Theory November 17, 2009 33 / 51
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 (Murdoch University) CoW & Number Theory November 17, 2009 33 / 51
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 (Murdoch University) CoW & Number Theory November 17, 2009 33 / 51
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 (Murdoch University) CoW & Number Theory November 17, 2009 33 / 51
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 (Murdoch University) CoW & Number Theory November 17, 2009 33 / 51
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 (Murdoch University) CoW & Number Theory November 17, 2009 33 / 51
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 (Murdoch University) CoW & Number Theory November 17, 2009 33 / 51
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 (Murdoch University) CoW & Number Theory November 17, 2009 33 / 51
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 (Murdoch University) CoW & Number Theory November 17, 2009 33 / 51
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 (Murdoch University) CoW & Number Theory November 17, 2009 33 / 51
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 (Murdoch University) CoW & Number Theory November 17, 2009 33 / 51
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 Tribonaccinumbers 1, 1, 2, 4, 7, 13, 24, 44, . . . , defined by:
T0 = T1 = 1, T2 = 2, Tn = Tn−1 + Tn−2 + Tn−3 for n ≥ 3.
Amy Glen (Murdoch University) CoW & Number Theory November 17, 2009 33 / 51
Combinatorics on Words Sturmian & Episturmian Words
A further generalisation: Rich words
Sturmian and episturmian words are “rich” in palindromes.
Amy Glen (Murdoch University) CoW & Number Theory November 17, 2009 34 / 51
Combinatorics on Words Sturmian & Episturmian Words
A further generalisation: Rich words
Sturmian and episturmian words are “rich” in palindromes.
Droubay, Justin, Pirillo (2001): Any finite word w of length |w |contains at most |w |+ 1 distinct palindromes (including ε).
Amy Glen (Murdoch University) CoW & Number Theory November 17, 2009 34 / 51
Combinatorics on Words Sturmian & Episturmian Words
A further generalisation: Rich words
Sturmian and episturmian words are “rich” in palindromes.
Droubay, Justin, Pirillo (2001): Any finite word w of length |w |contains at most |w |+ 1 distinct palindromes (including ε).
G.-Justin (2007): Initiated a unified study of finite and infinite wordsthat are characterized by containing the maximal number of distinctpalindromes, called rich words.
Amy Glen (Murdoch University) CoW & Number Theory November 17, 2009 34 / 51
Combinatorics on Words Sturmian & Episturmian Words
A further generalisation: Rich words
Sturmian and episturmian words are “rich” in palindromes.
Droubay, Justin, Pirillo (2001): Any finite word w of length |w |contains at most |w |+ 1 distinct palindromes (including ε).
G.-Justin (2007): Initiated a unified study of finite and infinite wordsthat are characterized by containing the maximal number of distinctpalindromes, called rich words.
A finite or infinite word w is rich if each factor u of w contains exactly|u|+ 1 distinct palindromic factors.
Amy Glen (Murdoch University) CoW & Number Theory November 17, 2009 34 / 51
Combinatorics on Words Sturmian & Episturmian Words
A further generalisation: Rich words
Sturmian and episturmian words are “rich” in palindromes.
Droubay, Justin, Pirillo (2001): Any finite word w of length |w |contains at most |w |+ 1 distinct palindromes (including ε).
G.-Justin (2007): Initiated a unified study of finite and infinite wordsthat are characterized by containing the maximal number of distinctpalindromes, called rich words.
A finite or infinite word w is rich if each factor u of w contains exactly|u|+ 1 distinct palindromic factors.
Characteristic property: All ‘complete returns’ to palindromes arepalindromes.
Amy Glen (Murdoch University) CoW & Number Theory November 17, 2009 34 / 51
Combinatorics on Words Sturmian & Episturmian Words
A further generalisation: Rich words
Sturmian and episturmian words are “rich” in palindromes.
Droubay, Justin, Pirillo (2001): Any finite word w of length |w |contains at most |w |+ 1 distinct palindromes (including ε).
G.-Justin (2007): Initiated a unified study of finite and infinite wordsthat are characterized by containing the maximal number of distinctpalindromes, called rich words.
A finite or infinite word w is rich if each factor u of w contains exactly|u|+ 1 distinct palindromic factors.
Characteristic property: All ‘complete returns’ to palindromes arepalindromes.
Examples: Sturmian and episturmian words
Amy Glen (Murdoch University) CoW & Number Theory November 17, 2009 34 / 51
Combinatorics on Words Sturmian & Episturmian Words
A further generalisation: Rich words
Sturmian and episturmian words are “rich” in palindromes.
Droubay, Justin, Pirillo (2001): Any finite word w of length |w |contains at most |w |+ 1 distinct palindromes (including ε).
G.-Justin (2007): Initiated a unified study of finite and infinite wordsthat are characterized by containing the maximal number of distinctpalindromes, called rich words.
A finite or infinite word w is rich if each factor u of w contains exactly|u|+ 1 distinct palindromic factors.
Characteristic property: All ‘complete returns’ to palindromes arepalindromes.
Examples: aaaaaa · · ·
Amy Glen (Murdoch University) CoW & Number Theory November 17, 2009 34 / 51
Combinatorics on Words Sturmian & Episturmian Words
A further generalisation: Rich words
Sturmian and episturmian words are “rich” in palindromes.
Droubay, Justin, Pirillo (2001): Any finite word w of length |w |contains at most |w |+ 1 distinct palindromes (including ε).
G.-Justin (2007): Initiated a unified study of finite and infinite wordsthat are characterized by containing the maximal number of distinctpalindromes, called rich words.
A finite or infinite word w is rich if each factor u of w contains exactly|u|+ 1 distinct palindromic factors.
Characteristic property: All ‘complete returns’ to palindromes arepalindromes.
Examples: abbbbbb · · ·
Amy Glen (Murdoch University) CoW & Number Theory November 17, 2009 34 / 51
Combinatorics on Words Sturmian & Episturmian Words
A further generalisation: Rich words
Sturmian and episturmian words are “rich” in palindromes.
Droubay, Justin, Pirillo (2001): Any finite word w of length |w |contains at most |w |+ 1 distinct palindromes (including ε).
G.-Justin (2007): Initiated a unified study of finite and infinite wordsthat are characterized by containing the maximal number of distinctpalindromes, called rich words.
A finite or infinite word w is rich if each factor u of w contains exactly|u|+ 1 distinct palindromic factors.
Characteristic property: All ‘complete returns’ to palindromes arepalindromes.
Examples: abaabaaabaaaab · · ·
Amy Glen (Murdoch University) CoW & Number Theory November 17, 2009 34 / 51
Combinatorics on Words Sturmian & Episturmian Words
A further generalisation: Rich words
Sturmian and episturmian words are “rich” in palindromes.
Droubay, Justin, Pirillo (2001): Any finite word w of length |w |contains at most |w |+ 1 distinct palindromes (including ε).
G.-Justin (2007): Initiated a unified study of finite and infinite wordsthat are characterized by containing the maximal number of distinctpalindromes, called rich words.
A finite or infinite word w is rich if each factor u of w contains exactly|u|+ 1 distinct palindromic factors.
Characteristic property: All ‘complete returns’ to palindromes arepalindromes.
Examples: (abcba)(abcba)(abcba) · · ·
Amy Glen (Murdoch University) CoW & Number Theory November 17, 2009 34 / 51
Some Applications in Number Theory
Outline
1 Combinatorics on WordsSturmian & Episturmian Words
2 Some Applications in Number TheoryContinued Fractions & Sturmian WordsDistribution modulo 1 & Sturmian WordsTranscendental Numbers
Amy Glen (Murdoch University) CoW & Number Theory November 17, 2009 35 / 51
Some Applications in Number Theory Continued Fractions & Sturmian Words
Continued fractions
Every irrational number α > 0 has a unique continued fraction expansion
α = [a0; a1, a2, a3, . . .] = a0 +1
a1 +1
a2 +1
a3 + · · ·where the ai are non-negative integers, called partial quotients, with a0 ≥ 0& all other ai ≥ 1. The n-th convergent to α is the rational number:
pn
qn
= [a0; a1, a2, . . . , an], n ≥ 1.
Amy Glen (Murdoch University) CoW & Number Theory November 17, 2009 36 / 51
Some Applications in Number Theory Continued Fractions & Sturmian Words
Continued fractions
Every irrational number α > 0 has a unique continued fraction expansion
α = [a0; a1, a2, a3, . . .] = a0 +1
a1 +1
a2 +1
a3 + · · ·where the ai are non-negative integers, called partial quotients, with a0 ≥ 0& all other ai ≥ 1. The n-th convergent to α is the rational number:
pn
qn
= [a0; a1, a2, . . . , an], n ≥ 1.
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 (Murdoch University) CoW & Number Theory November 17, 2009 36 / 51
Some Applications in Number Theory Continued Fractions & Sturmian Words
Standard Sturmian words via CF expansions
Let cα denote the standard Sturmian word corresponding to the line ofirrational slope α ∈ (0, 1) through (0, 0) where
α = [0; a1, a2, a3, . . .].
Amy Glen (Murdoch University) CoW & Number Theory November 17, 2009 37 / 51
Some Applications in Number Theory Continued Fractions & Sturmian Words
Standard Sturmian words via CF expansions
Let cα denote the standard Sturmian word corresponding to the line ofirrational slope α ∈ (0, 1) through (0, 0) where
α = [0; a1, a2, a3, . . .].
Then cα = Pal(aa1ba2aa3ba4aa5 · · · ). [Arnoux-Rauzy 1991]
Amy Glen (Murdoch University) CoW & Number Theory November 17, 2009 37 / 51
Some Applications in Number Theory Continued Fractions & Sturmian Words
Standard Sturmian words via CF expansions
Let cα denote the standard Sturmian word corresponding to the line ofirrational slope α ∈ (0, 1) through (0, 0) where
α = [0; a1, a2, a3, . . .].
Then cα = Pal(aa1ba2aa3ba4aa5 · · · ). [Arnoux-Rauzy 1991]
Example: Recall the Fibonacci word is f := cα withα = 1/τ = (
√5− 1)/2 = [0; 1, 1, 1, 1, . . .] and combinatorially:
f = Pal(abababab · · · ) = abaababaaba · · · .
Amy Glen (Murdoch University) CoW & Number Theory November 17, 2009 37 / 51
Some Applications in Number Theory Continued Fractions & Sturmian Words
Standard Sturmian words via CF expansions
Let cα denote the standard Sturmian word corresponding to the line ofirrational slope α ∈ (0, 1) through (0, 0) where
α = [0; a1, a2, a3, . . .].
Then cα = Pal(aa1ba2aa3ba4aa5 · · · ). [Arnoux-Rauzy 1991]
Example: Recall the Fibonacci word is f := cα withα = 1/τ = (
√5− 1)/2 = [0; 1, 1, 1, 1, . . .] and combinatorially:
f = Pal(abababab · · · ) = abaababaaba · · · .
cα can also be constructed as the limit of an infinite sequence of finitewords, defined with respect to the CF of α.
Amy Glen (Murdoch University) CoW & Number Theory November 17, 2009 37 / 51
Some Applications in Number Theory Continued Fractions & Sturmian Words
Standard Sturmian words via CF expansions
Let cα denote the standard Sturmian word corresponding to the line ofirrational slope α ∈ (0, 1) through (0, 0) where
α = [0; a1, a2, a3, . . .].
Then cα = Pal(aa1ba2aa3ba4aa5 · · · ). [Arnoux-Rauzy 1991]
Example: Recall the Fibonacci word is f := cα withα = 1/τ = (
√5− 1)/2 = [0; 1, 1, 1, 1, . . .] and combinatorially:
f = Pal(abababab · · · ) = abaababaaba · · · .
cα can also be constructed as the limit of an infinite sequence of finitewords, defined with respect to the CF of α.
Many nice combinatorial properties of cα are related to the CF of α.
Amy Glen (Murdoch University) CoW & Number Theory November 17, 2009 37 / 51
Some Applications in Number Theory Distribution modulo 1 & Sturmian Words
Fractional parts of powers & Sturmian words
Mahler (1968): defined the set of Z -numbers by
Z :=
{
ξ ∈ R, ξ > 0, ∀n ≥ 0, 0 ≤{
ξ
(3
2
)n}
<1
2
}
where {z} denotes the fractional part of z .
Amy Glen (Murdoch University) CoW & Number Theory November 17, 2009 38 / 51
Some Applications in Number Theory Distribution modulo 1 & Sturmian Words
Fractional parts of powers & Sturmian words
Mahler (1968): defined the set of Z -numbers by
Z :=
{
ξ ∈ R, ξ > 0, ∀n ≥ 0, 0 ≤{
ξ
(3
2
)n}
<1
2
}
where {z} denotes the fractional part of z .
Mahler proved that Z is at most countable.
Amy Glen (Murdoch University) CoW & Number Theory November 17, 2009 38 / 51
Some Applications in Number Theory Distribution modulo 1 & Sturmian Words
Fractional parts of powers & Sturmian words
Mahler (1968): defined the set of Z -numbers by
Z :=
{
ξ ∈ R, ξ > 0, ∀n ≥ 0, 0 ≤{
ξ
(3
2
)n}
<1
2
}
where {z} denotes the fractional part of z .
Mahler proved that Z is at most countable.
It is still an open problem to prove that Z = ∅.
Amy Glen (Murdoch University) CoW & Number Theory November 17, 2009 38 / 51
Some Applications in Number Theory Distribution modulo 1 & Sturmian Words
Fractional parts of powers & Sturmian words
Mahler (1968): defined the set of Z -numbers by
Z :=
{
ξ ∈ R, ξ > 0, ∀n ≥ 0, 0 ≤{
ξ
(3
2
)n}
<1
2
}
where {z} denotes the fractional part of z .
Mahler proved that Z is at most countable.
It is still an open problem to prove that Z = ∅.
More general question:
Given a real number α > 1 and an interval (x , y) ⊂ (0, 1),does there exists ξ > 0 such that, for all n ≥ 0, we havex ≤ {ξαn} < y (or the variant x ≤ {ξαn} ≤ y)?
Amy Glen (Murdoch University) CoW & Number Theory November 17, 2009 38 / 51
Some Applications in Number Theory Distribution modulo 1 & Sturmian Words
Fractional parts of powers & Sturmian words . . .
Flatto-Lagarias-Pollington, 1995
If p, q are coprime integers and p > q ≥ 2, then any interval (x , y) suchthat for some ξ > 0, one has that {ξ(p/q)n} ∈ (x , y) for all n ≥ 0, mustsatisfy y − x ≥ 1/p.
Amy Glen (Murdoch University) CoW & Number Theory November 17, 2009 39 / 51
Some Applications in Number Theory Distribution modulo 1 & Sturmian Words
Fractional parts of powers & Sturmian words . . .
Flatto-Lagarias-Pollington, 1995
If p, q are coprime integers and p > q ≥ 2, then any interval (x , y) suchthat for some ξ > 0, one has that {ξ(p/q)n} ∈ (x , y) for all n ≥ 0, mustsatisfy y − x ≥ 1/p.
Bugeaud-Dubickas (2005): described all irrational numbers ξ > 0 such thatfor a fixed integer b ≥ 2 the fractional parts {ξbn}, n ≥ 0, all belong to aninterval of length 1/b.
Amy Glen (Murdoch University) CoW & Number Theory November 17, 2009 39 / 51
Some Applications in Number Theory Distribution modulo 1 & Sturmian Words
Fractional parts of powers & Sturmian words . . .
Flatto-Lagarias-Pollington, 1995
If p, q are coprime integers and p > q ≥ 2, then any interval (x , y) suchthat for some ξ > 0, one has that {ξ(p/q)n} ∈ (x , y) for all n ≥ 0, mustsatisfy y − x ≥ 1/p.
Bugeaud-Dubickas (2005): described all irrational numbers ξ > 0 such thatfor a fixed integer b ≥ 2 the fractional parts {ξbn}, n ≥ 0, all belong to aninterval of length 1/b.
Bugeaud-Dubickas, 2005
Let b ≥ 2 be an integer and ξ > 0 be an irrational number. Then:
the numbers {ξbn} cannot all lie in an interval of length < 1/b;
there exists a closed interval of length 1/b containing the numbers{ξbn} for all n ≥ 0 iff the base b expansion of the fractional part of ξis a Sturmian sequence on {k , k + 1} for some k ∈ {0, 1, . . . , b− 2}.
Amy Glen (Murdoch University) CoW & Number Theory November 17, 2009 39 / 51
Some Applications in Number Theory Distribution modulo 1 & Sturmian Words
Fractional parts of powers & Sturmian words . . .
The core of Bugeaud & Dubickas’ result is the following:
Theorem
An aperiodic sequence s := (sn)n≥0 on {0, 1} is Sturmian if and only ifthere exists a sequence u := (un)n≥0 on {0, 1} such that
0u ≤ T k(s) ≤ 1u for all k ≥ 0,
where T k denotes the k-th iterate of the shift map T defined byT ((sn)n≥1) = (sn+1)n≥1 and ≤ is the lexicographic order on sequencesover {0, 1} induced by 0 < 1.
Moreover, u is the unique standard Sturmian sequence with the same slopeas s , and we have
0u = inf{T k(s), k ≥ 0} and 1u = sup{T k(s), k ≥ 0}.
Amy Glen (Murdoch University) CoW & Number Theory November 17, 2009 40 / 51
Some Applications in Number Theory Distribution modulo 1 & Sturmian Words
Fractional parts of powers & Sturmian words . . .
In particular, all shifts of a Sturmian sequence s of slope α over {0, 1}are lexicographically ≥ 0cα and lexicographically ≤ 1cα.
Amy Glen (Murdoch University) CoW & Number Theory November 17, 2009 41 / 51
Some Applications in Number Theory Distribution modulo 1 & Sturmian Words
Fractional parts of powers & Sturmian words . . .
In particular, all shifts of a Sturmian sequence s of slope α over {0, 1}are lexicographically ≥ 0cα and lexicographically ≤ 1cα.
Example: Consider the Fibonacci word on {0, 1}:f = 0100101001001010010 · · · ,
the standard Sturmian word of slope α = (√
5− 1)/2.
Amy Glen (Murdoch University) CoW & Number Theory November 17, 2009 41 / 51
Some Applications in Number Theory Distribution modulo 1 & Sturmian Words
Fractional parts of powers & Sturmian words . . .
In particular, all shifts of a Sturmian sequence s of slope α over {0, 1}are lexicographically ≥ 0cα and lexicographically ≤ 1cα.
Example: Consider the Fibonacci word on {0, 1}:f = 0100101001001010010 · · · ,
the standard Sturmian word of slope α = (√
5− 1)/2.
001001010010010 · · ·︸ ︷︷ ︸
0f
<
f︷ ︸︸ ︷
01001010010010 · · · < 101001010010010 · · ·︸ ︷︷ ︸
1f
Amy Glen (Murdoch University) CoW & Number Theory November 17, 2009 41 / 51
Some Applications in Number Theory Distribution modulo 1 & Sturmian Words
Fractional parts of powers & Sturmian words . . .
In particular, all shifts of a Sturmian sequence s of slope α over {0, 1}are lexicographically ≥ 0cα and lexicographically ≤ 1cα.
Example: Consider the Fibonacci word on {0, 1}:f = 0100101001001010010 · · · ,
the standard Sturmian word of slope α = (√
5− 1)/2.
001001010010010 · · ·︸ ︷︷ ︸
0f
<
T (f )︷ ︸︸ ︷
1001010010010 · · · < 101001010010010 · · ·︸ ︷︷ ︸
1f
Amy Glen (Murdoch University) CoW & Number Theory November 17, 2009 41 / 51
Some Applications in Number Theory Distribution modulo 1 & Sturmian Words
Fractional parts of powers & Sturmian words . . .
In particular, all shifts of a Sturmian sequence s of slope α over {0, 1}are lexicographically ≥ 0cα and lexicographically ≤ 1cα.
Example: Consider the Fibonacci word on {0, 1}:f = 0100101001001010010 · · · ,
the standard Sturmian word of slope α = (√
5− 1)/2.
001001010010010 · · ·︸ ︷︷ ︸
0f
<
T2(f )
︷ ︸︸ ︷
001010010010 · · · < 101001010010010 · · ·︸ ︷︷ ︸
1f
Amy Glen (Murdoch University) CoW & Number Theory November 17, 2009 41 / 51
Some Applications in Number Theory Distribution modulo 1 & Sturmian Words
Fractional parts of powers & Sturmian words . . .
In particular, all shifts of a Sturmian sequence s of slope α over {0, 1}are lexicographically ≥ 0cα and lexicographically ≤ 1cα.
Example: Consider the Fibonacci word on {0, 1}:f = 0100101001001010010 · · · ,
the standard Sturmian word of slope α = (√
5− 1)/2.
001001010010010 · · ·︸ ︷︷ ︸
0f
<
T3(f )
︷ ︸︸ ︷
01010010010 · · · < 101001010010010 · · ·︸ ︷︷ ︸
1f
Amy Glen (Murdoch University) CoW & Number Theory November 17, 2009 41 / 51
Some Applications in Number Theory Distribution modulo 1 & Sturmian Words
Fractional parts of powers & Sturmian words . . .
In particular, all shifts of a Sturmian sequence s of slope α over {0, 1}are lexicographically ≥ 0cα and lexicographically ≤ 1cα.
Example: Consider the Fibonacci word on {0, 1}:f = 0100101001001010010 · · · ,
the standard Sturmian word of slope α = (√
5− 1)/2.
001001010010010 · · ·︸ ︷︷ ︸
0f
<
T4(f )
︷ ︸︸ ︷
1010010010 · · · < 101001010010010 · · ·︸ ︷︷ ︸
1f
and so on . . .
Amy Glen (Murdoch University) CoW & Number Theory November 17, 2009 41 / 51
Some Applications in Number Theory Distribution modulo 1 & Sturmian Words
Fractional parts of powers & Sturmian words . . .
In particular, all shifts of a Sturmian sequence s of slope α over {0, 1}are lexicographically ≥ 0cα and lexicographically ≤ 1cα.
Example: Consider the Fibonacci word on {0, 1}:f = 0100101001001010010 · · · ,
the standard Sturmian word of slope α = (√
5− 1)/2.
001001010010010 · · ·︸ ︷︷ ︸
0f
<
T4(f )
︷ ︸︸ ︷
1010010010 · · · < 101001010010010 · · ·︸ ︷︷ ︸
1f
and so on . . .
The preceding theorem has been rediscovered numerous times sincethe work of P. Veerman in the mid-late 80’s (which was the first timeit was proved as far as we can ascertain).
Amy Glen (Murdoch University) CoW & Number Theory November 17, 2009 41 / 51
Some Applications in Number Theory Distribution modulo 1 & Sturmian Words
The main tool used by Bugeaud and Dubickas was combinatorics on
words: replace real numbers by their base b expansions, and transforminequalities between real numbers into (lexicographic) inequalitiesbetween infinite sequences representing their base b expansions.
Amy Glen (Murdoch University) CoW & Number Theory November 17, 2009 42 / 51
Some Applications in Number Theory Distribution modulo 1 & Sturmian Words
The main tool used by Bugeaud and Dubickas was combinatorics on
words: replace real numbers by their base b expansions, and transforminequalities between real numbers into (lexicographic) inequalitiesbetween infinite sequences representing their base b expansions.
Given a sequence s over {0, 1}, let r(s) denote the real number whosebase 2 expansion is s .
Now let x , y , s be sequences over {0, 1}. Then
x ≤ T k(s) ≤ y ←→ r(x) ≤ {r(s)2k} ≤ r(y).
Amy Glen (Murdoch University) CoW & Number Theory November 17, 2009 42 / 51
Some Applications in Number Theory Distribution modulo 1 & Sturmian Words
The main tool used by Bugeaud and Dubickas was combinatorics on
words: replace real numbers by their base b expansions, and transforminequalities between real numbers into (lexicographic) inequalitiesbetween infinite sequences representing their base b expansions.
Given a sequence s over {0, 1}, let r(s) denote the real number whosebase 2 expansion is s .
Now let x , y , s be sequences over {0, 1}. Then
x ≤ T k(s) ≤ y ←→ r(x) ≤ {r(s)2k} ≤ r(y).
Example:
(01)∞ ≤ T k((01)∞) ≤ (10)∞ for all k ≥ 0
13 ≤ {1
3 · 2k} ≤ 23 for all k ≥ 0
Amy Glen (Murdoch University) CoW & Number Theory November 17, 2009 42 / 51
Some Applications in Number Theory Distribution modulo 1 & Sturmian Words
The main tool used by Bugeaud and Dubickas was combinatorics on
words: replace real numbers by their base b expansions, and transforminequalities between real numbers into (lexicographic) inequalitiesbetween infinite sequences representing their base b expansions.
Given a sequence s over {0, 1}, let r(s) denote the real number whosebase 2 expansion is s .
Now let x , y , s be sequences over {0, 1}. Then
x ≤ T k(s) ≤ y ←→ r(x) ≤ {r(s)2k} ≤ r(y).
Example:
(01)∞ ≤ T k((01)∞) ≤ (10)∞ for all k ≥ 0
13 ≤ {1
3 · 2k} ≤ 23 for all k ≥ 0
We have obtained a complete description of the minimal intervalscontaining all fractional parts {ξ2n}, n ≥ 0, for some real number ξ > 0 . . .Amy Glen (Murdoch University) CoW & Number Theory November 17, 2009 42 / 51
Some Applications in Number Theory Distribution modulo 1 & Sturmian Words
Definition
For all x ∈ [0, 1], let Sx := {ξ ∈ R, ξ > 0, ∀n ≥ 0, x ≤ {ξ2n} < 1} andlet F : [0, 1]→ [0, 1] be the function defined by:
F (x) =
{
inf{y ∈ [0, 1), ∃ξ > 0, ∀n ≥ 0, x ≤ {ξ2n} ≤ y} if Sx 6= ∅,1 if Sx = ∅.
Amy Glen (Murdoch University) CoW & Number Theory November 17, 2009 43 / 51
Some Applications in Number Theory Distribution modulo 1 & Sturmian Words
Definition
For all x ∈ [0, 1], let Sx := {ξ ∈ R, ξ > 0, ∀n ≥ 0, x ≤ {ξ2n} < 1} andlet F : [0, 1]→ [0, 1] be the function defined by:
F (x) =
{
inf{y ∈ [0, 1), ∃ξ > 0, ∀n ≥ 0, x ≤ {ξ2n} ≤ y} if Sx 6= ∅,1 if Sx = ∅.
From Bugeaud & Dubickas’ result for b = 2, we deduce the following facts.
For x ∈ [12 , 1], there does not exist a real number ξ > 0 such thatx ≤ {ξ2n} < 1 for all n ≥ 0.
Amy Glen (Murdoch University) CoW & Number Theory November 17, 2009 43 / 51
Some Applications in Number Theory Distribution modulo 1 & Sturmian Words
Definition
For all x ∈ [0, 1], let Sx := {ξ ∈ R, ξ > 0, ∀n ≥ 0, x ≤ {ξ2n} < 1} andlet F : [0, 1]→ [0, 1] be the function defined by:
F (x) =
{
inf{y ∈ [0, 1), ∃ξ > 0, ∀n ≥ 0, x ≤ {ξ2n} ≤ y} if Sx 6= ∅,1 if Sx = ∅.
From Bugeaud & Dubickas’ result for b = 2, we deduce the following facts.
For x ∈ [12 , 1], there does not exist a real number ξ > 0 such thatx ≤ {ξ2n} < 1 for all n ≥ 0.
Hence, F (x) = 1 for all x ∈ [12 , 1].
Amy Glen (Murdoch University) CoW & Number Theory November 17, 2009 43 / 51
Some Applications in Number Theory Distribution modulo 1 & Sturmian Words
Definition
For all x ∈ [0, 1], let Sx := {ξ ∈ R, ξ > 0, ∀n ≥ 0, x ≤ {ξ2n} < 1} andlet F : [0, 1]→ [0, 1] be the function defined by:
F (x) =
{
inf{y ∈ [0, 1), ∃ξ > 0, ∀n ≥ 0, x ≤ {ξ2n} ≤ y} if Sx 6= ∅,1 if Sx = ∅.
From Bugeaud & Dubickas’ result for b = 2, we deduce the following facts.
For x ∈ [12 , 1], there does not exist a real number ξ > 0 such thatx ≤ {ξ2n} < 1 for all n ≥ 0.
Hence, F (x) = 1 for all x ∈ [12 , 1].
If ξ > 0 is an irrational real number, then there exists a real numberx ∈ [0, 1
2 ) such that all the fractional parts {ξ2n} belong to theinterval [x , x + 1
2 ] if and only if the base 2 expansion of the fractionalpart of ξ is a Sturmian sequence.
Amy Glen (Murdoch University) CoW & Number Theory November 17, 2009 43 / 51
Some Applications in Number Theory Distribution modulo 1 & Sturmian Words
Definition
For all x ∈ [0, 1], let Sx := {ξ ∈ R, ξ > 0, ∀n ≥ 0, x ≤ {ξ2n} < 1} andlet F : [0, 1]→ [0, 1] be the function defined by:
F (x) =
{
inf{y ∈ [0, 1), ∃ξ > 0, ∀n ≥ 0, x ≤ {ξ2n} ≤ y} if Sx 6= ∅,1 if Sx = ∅.
From Bugeaud & Dubickas’ result for b = 2, we deduce the following facts.
For x ∈ [12 , 1], there does not exist a real number ξ > 0 such thatx ≤ {ξ2n} < 1 for all n ≥ 0.
Hence, F (x) = 1 for all x ∈ [12 , 1].
If ξ > 0 is an irrational real number, then there exists a real numberx ∈ [0, 1
2 ) such that all the fractional parts {ξ2n} belong to theinterval [x , x + 1
2 ] if and only if the base 2 expansion of the fractionalpart of ξ is a Sturmian sequence.
Furthermore, for any such x , one has F (x) = x + 12 .
Amy Glen (Murdoch University) CoW & Number Theory November 17, 2009 43 / 51
Some Applications in Number Theory Distribution modulo 1 & Sturmian Words
Theorem (Allouche-G., 2009)
Let x be a real number in [0, 1].
(i) If x ≥ 12 , then F (x) = 1.
Amy Glen (Murdoch University) CoW & Number Theory November 17, 2009 44 / 51
Some Applications in Number Theory Distribution modulo 1 & Sturmian Words
Theorem (Allouche-G., 2009)
Let x be a real number in [0, 1].
(i) If x ≥ 12 , then F (x) = 1.
(ii) If x = 0, then F (x) = 0.
Amy Glen (Murdoch University) CoW & Number Theory November 17, 2009 44 / 51
Some Applications in Number Theory Distribution modulo 1 & Sturmian Words
Theorem (Allouche-G., 2009)
Let x be a real number in [0, 1].
(i) If x ≥ 12 , then F (x) = 1.
(ii) If x = 0, then F (x) = 0.
(iii) If x ∈ (0, 12) and if the base 2 expansion of 2x is given by a standard
Sturmian sequence, then F (x) = x + 12 .
Furthermore, F (x) is the unique real number in [0, 1] that has aSturmian base 2 expansion and satisfies x ≤ {F (x)2k} ≤ F (x) for allk ≥ 0.
Amy Glen (Murdoch University) CoW & Number Theory November 17, 2009 44 / 51
Some Applications in Number Theory Distribution modulo 1 & Sturmian Words
Theorem (Allouche-G., 2009)
Let x be a real number in [0, 1].
(i) If x ≥ 12 , then F (x) = 1.
(ii) If x = 0, then F (x) = 0.
(iii) If x ∈ (0, 12) and if the base 2 expansion of 2x is given by a standard
Sturmian sequence, then F (x) = x + 12 .
Furthermore, F (x) is the unique real number in [0, 1] that has aSturmian base 2 expansion and satisfies x ≤ {F (x)2k} ≤ F (x) for allk ≥ 0.
Let Sα denote the set of all Sturmian sequences of (irrational) slopeα ∈ (0, 1) over the alphabet {0, 1} (i.e., a 7→ 0, b 7→ 1).
Noting that r(1cα) = 1/2 + r(0cα), we have that the setr(Sα) := {r(s) ∈ [0, 1), s ∈ Sα} is completely contained within theclosed interval [r(0cα), r(1cα)] of length 1/2 (and no smaller).
Amy Glen (Murdoch University) CoW & Number Theory November 17, 2009 44 / 51
Some Applications in Number Theory Distribution modulo 1 & Sturmian Words
Theorem (Allouche-G., 2009) . . .
(iv) If x ∈ (0, 12) and if the base 2 expansion of 2x is a periodic sequence
of the form (Pal(v)01)∞ or (Pal(v)10)∞ for some v ∈ {0, 1}∗, thenF (x) is the rational number whose base 2 expansion is the periodicsequence (1Pal(v)0)∞, in which case F (x) ≤ x + 1
2 .
Amy Glen (Murdoch University) CoW & Number Theory November 17, 2009 45 / 51
Some Applications in Number Theory Distribution modulo 1 & Sturmian Words
Theorem (Allouche-G., 2009) . . .
(iv) If x ∈ (0, 12) and if the base 2 expansion of 2x is a periodic sequence
of the form (Pal(v)01)∞ or (Pal(v)10)∞ for some v ∈ {0, 1}∗, thenF (x) is the rational number whose base 2 expansion is the periodicsequence (1Pal(v)0)∞, in which case F (x) ≤ x + 1
2 .
(v) In all other cases, F (x) can be explicitly computed: it is equal to therational number whose base 2 expansion is a periodic sequence of theform (1Pal(v)0)∞ for some (unique) v ∈ {0, 1}∗ such that
r((Pal(v)01)∞) < 2x < r((Pal(v)10)∞).
In these cases, F (x) < x + 12 ·
Amy Glen (Murdoch University) CoW & Number Theory November 17, 2009 45 / 51
Some Applications in Number Theory Distribution modulo 1 & Sturmian Words
Theorem (Allouche-G., 2009) . . .
(iv) If x ∈ (0, 12) and if the base 2 expansion of 2x is a periodic sequence
of the form (Pal(v)01)∞ or (Pal(v)10)∞ for some v ∈ {0, 1}∗, thenF (x) is the rational number whose base 2 expansion is the periodicsequence (1Pal(v)0)∞, in which case F (x) ≤ x + 1
2 .
(v) In all other cases, F (x) can be explicitly computed: it is equal to therational number whose base 2 expansion is a periodic sequence of theform (1Pal(v)0)∞ for some (unique) v ∈ {0, 1}∗ such that
r((Pal(v)01)∞) < 2x < r((Pal(v)10)∞).
In these cases, F (x) < x + 12 ·
Moreover, in cases (iv) and (v), F (x) is the unique real number in (0, 1)whose base 2 expansion is a periodic sequence of the form (1Pal(v)0)∞
and which satisfies x ≤ {F (x)2k} ≤ F (x) for all k ≥ 0.Amy Glen (Murdoch University) CoW & Number Theory November 17, 2009 45 / 51
Some Applications in Number Theory Distribution modulo 1 & Sturmian Words
Examples
F (1/4) = 2/3 since the base 2 expansion of 1/4 is
(1/4)2 = 01000 · · · (or 00111 · · · ),and we have
(01)∞ < (2× (1/4))2 (= 1000 · · · ) < (10)∞
where (10)∞ is the base 2 expansion of 2/3. (Here Pal(v) = ε.)
Amy Glen (Murdoch University) CoW & Number Theory November 17, 2009 46 / 51
Some Applications in Number Theory Distribution modulo 1 & Sturmian Words
Examples
F (1/4) = 2/3 since the base 2 expansion of 1/4 is
(1/4)2 = 01000 · · · (or 00111 · · · ),and we have
(01)∞ < (2× (1/4))2 (= 1000 · · · ) < (10)∞
where (10)∞ is the base 2 expansion of 2/3. (Here Pal(v) = ε.)
F (1/2π) = 20/31 (< 1/2π + 1/2 = 0.65915 . . .) since the base 2expansion of 1/2π is
(1/2π)2 = 0 01010︸ ︷︷ ︸
Pal(011)
00101111100110000011011011100 · · ·
and we have
(01001)∞ < (2× (1/2π))2 < (01010)∞
where (10100)∞ is the base 2 expansion of 20/31.
Amy Glen (Murdoch University) CoW & Number Theory November 17, 2009 46 / 51
Some Applications in Number Theory Distribution modulo 1 & Sturmian Words
Remarks
It is known (Ferenczi-Mauduit, 1997) that real numbers having aSturmian base 2 expansion are transcendental.
As a consequence of our theorem, we deduce:
If x is an algebraic real number in [0, 12), then F (x) is rational.
Amy Glen (Murdoch University) CoW & Number Theory November 17, 2009 47 / 51
Some Applications in Number Theory Distribution modulo 1 & Sturmian Words
Remarks
It is known (Ferenczi-Mauduit, 1997) that real numbers having aSturmian base 2 expansion are transcendental.
As a consequence of our theorem, we deduce:
If x is an algebraic real number in [0, 12), then F (x) is rational.
One may ask what happens with base b expansions, where b ≥ 3, orwhat can be said about the intervals containing all {ξbn} for some ξ.
Amy Glen (Murdoch University) CoW & Number Theory November 17, 2009 47 / 51
Some Applications in Number Theory Distribution modulo 1 & Sturmian Words
Remarks
It is known (Ferenczi-Mauduit, 1997) that real numbers having aSturmian base 2 expansion are transcendental.
As a consequence of our theorem, we deduce:
If x is an algebraic real number in [0, 12), then F (x) is rational.
One may ask what happens with base b expansions, where b ≥ 3, orwhat can be said about the intervals containing all {ξbn} for some ξ.
The result of Bugeaud and Dubickas (2005) implies that Sturmiansequences on an alphabet {k , k + 1} for some k ∈ {0, 1, . . . , b − 2}will again play a fundamental role.
Amy Glen (Murdoch University) CoW & Number Theory November 17, 2009 47 / 51
Some Applications in Number Theory Transcendental Numbers
Transcendental continued fractions
Long-standing Conjecture (Khintchine 1949)
The CF expansion of an irrational algebraic real number α is eithereventually periodic (iff α is a quadratic irrational) or it contains arbitrarilylarge partial quotients.
Amy Glen (Murdoch University) CoW & Number Theory November 17, 2009 48 / 51
Some Applications in Number Theory Transcendental Numbers
Transcendental continued fractions
Long-standing Conjecture (Khintchine 1949)
The CF expansion of an irrational algebraic real number α is eithereventually periodic (iff α is a quadratic irrational) or it contains arbitrarilylarge partial quotients.
Alternatively: An irrational number whose CF expansion has boundedpartial quotients is either quadratic or transcendental.
Amy Glen (Murdoch University) CoW & Number Theory November 17, 2009 48 / 51
Some Applications in Number Theory Transcendental Numbers
Transcendental continued fractions
Long-standing Conjecture (Khintchine 1949)
The CF expansion of an irrational algebraic real number α is eithereventually periodic (iff α is a quadratic irrational) or it contains arbitrarilylarge partial quotients.
Alternatively: An irrational number whose CF expansion has boundedpartial quotients is either quadratic or transcendental.
Liouville (1844): Transcendental CF’s whose sequences of partialquotients grow very fast (too fast to be algebraic)
Amy Glen (Murdoch University) CoW & Number Theory November 17, 2009 48 / 51
Some Applications in Number Theory Transcendental Numbers
Transcendental continued fractions
Long-standing Conjecture (Khintchine 1949)
The CF expansion of an irrational algebraic real number α is eithereventually periodic (iff α is a quadratic irrational) or it contains arbitrarilylarge partial quotients.
Alternatively: An irrational number whose CF expansion has boundedpartial quotients is either quadratic or transcendental.
Liouville (1844): Transcendental CF’s whose sequences of partialquotients grow very fast (too fast to be algebraic)
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 (Murdoch University) CoW & Number Theory November 17, 2009 48 / 51
Some Applications in 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 (Murdoch University) CoW & Number Theory November 17, 2009 49 / 51
Some Applications in 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 (Murdoch University) CoW & Number Theory November 17, 2009 49 / 51
Some Applications in 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 (Murdoch University) CoW & Number Theory November 17, 2009 49 / 51
Some Applications in 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 (Murdoch University) CoW & Number Theory November 17, 2009 49 / 51
Some Applications in 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 (Murdoch University) CoW & Number Theory November 17, 2009 49 / 51
Some Applications in 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 (Murdoch University) CoW & Number Theory November 17, 2009 49 / 51
Some Applications in 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 (Murdoch University) CoW & Number Theory November 17, 2009 49 / 51
Some Applications in 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 (Murdoch University) CoW & Number Theory November 17, 2009 49 / 51
Some Applications in 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 (Murdoch University) CoW & Number Theory November 17, 2009 49 / 51
Some Applications in 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 (Murdoch University) CoW & Number Theory November 17, 2009 50 / 51
Some Applications in 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 (Murdoch University) CoW & Number Theory November 17, 2009 50 / 51
Some Applications in 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 (Murdoch University) CoW & Number Theory November 17, 2009 50 / 51
Some Applications in 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 (Murdoch University) CoW & Number Theory November 17, 2009 50 / 51
Thank You!
Amy Glen (Murdoch University) CoW & Number Theory November 17, 2009 51 / 51
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