T HE ORIGIN OF COMBINATORICS ON WORDS Thue proved the following: ◮ There is an infinite word over the binary alphabet Σ 2 = { 0 , 1 } that contains no cubes as factors. Axel Thue (1863-1922)
T HE ORIGIN OF COMBINATORICS ON WORDS Thue proved the following: ◮ There is an infinite word over the binary alphabet Σ 2 = { 0 , 1 } that contains no cubes as factors. ◮ There is an infinite word over the ternary alphabet Σ 3 = { 0 , 1 , 2 } that contains no squares as factors. Axel Thue (1863-1922)
T HE ORIGIN OF COMBINATORICS ON WORDS Thue proved the following: ◮ There is an infinite word over the binary alphabet Σ 2 = { 0 , 1 } that contains no cubes as factors. ◮ There is an infinite word over the ternary alphabet Σ 3 = { 0 , 1 , 2 } that contains no squares as factors. ◮ There is no such word over Σ 2 . Axel Thue (1863-1922)
M ORPHIC WORDS Thue’s constructions relied on iterating a morphism .
M ORPHIC WORDS Thue’s constructions relied on iterating a morphism . ◮ Define a map µ by µ ( 0 ) = 01 and µ ( 1 ) = 10 .
M ORPHIC WORDS Thue’s constructions relied on iterating a morphism . ◮ Define a map µ by µ ( 0 ) = 01 and µ ( 1 ) = 10 . ◮ Extend µ to all words over { 0 , 1 } in the obvious way: µ ( 010 ) = µ ( 0 ) µ ( 1 ) µ ( 0 ) = 011001
M ORPHIC WORDS Thue’s constructions relied on iterating a morphism . ◮ Define a map µ by µ ( 0 ) = 01 and µ ( 1 ) = 10 . ◮ Extend µ to all words over { 0 , 1 } in the obvious way: µ ( 010 ) = µ ( 0 ) µ ( 1 ) µ ( 0 ) = 011001 To construct an infinite word: ◮ Start with 0 , and repeatedly apply µ .
M ORPHIC WORDS Thue’s constructions relied on iterating a morphism . ◮ Define a map µ by µ ( 0 ) = 01 and µ ( 1 ) = 10 . ◮ Extend µ to all words over { 0 , 1 } in the obvious way: µ ( 010 ) = µ ( 0 ) µ ( 1 ) µ ( 0 ) = 011001 To construct an infinite word: ◮ Start with 0 , and repeatedly apply µ . µ ( 0 ) = 01
M ORPHIC WORDS Thue’s constructions relied on iterating a morphism . ◮ Define a map µ by µ ( 0 ) = 01 and µ ( 1 ) = 10 . ◮ Extend µ to all words over { 0 , 1 } in the obvious way: µ ( 010 ) = µ ( 0 ) µ ( 1 ) µ ( 0 ) = 011001 To construct an infinite word: ◮ Start with 0 , and repeatedly apply µ . µ ( 0 ) = 01 µ 2 ( 0 ) = 0110
M ORPHIC WORDS Thue’s constructions relied on iterating a morphism . ◮ Define a map µ by µ ( 0 ) = 01 and µ ( 1 ) = 10 . ◮ Extend µ to all words over { 0 , 1 } in the obvious way: µ ( 010 ) = µ ( 0 ) µ ( 1 ) µ ( 0 ) = 011001 To construct an infinite word: ◮ Start with 0 , and repeatedly apply µ . µ ( 0 ) = 01 µ 2 ( 0 ) = 0110 µ 3 ( 0 ) = 01101001
M ORPHIC WORDS Thue’s constructions relied on iterating a morphism . ◮ Define a map µ by µ ( 0 ) = 01 and µ ( 1 ) = 10 . ◮ Extend µ to all words over { 0 , 1 } in the obvious way: µ ( 010 ) = µ ( 0 ) µ ( 1 ) µ ( 0 ) = 011001 To construct an infinite word: ◮ Start with 0 , and repeatedly apply µ . µ ( 0 ) = 01 µ 2 ( 0 ) = 0110 µ 3 ( 0 ) = 01101001 µ 4 ( 0 ) = 0110100110010110
M ORPHIC WORDS Thue’s constructions relied on iterating a morphism . ◮ Define a map µ by µ ( 0 ) = 01 and µ ( 1 ) = 10 . ◮ Extend µ to all words over { 0 , 1 } in the obvious way: µ ( 010 ) = µ ( 0 ) µ ( 1 ) µ ( 0 ) = 011001 To construct an infinite word: ◮ Start with 0 , and repeatedly apply µ . µ ( 0 ) = 01 µ 2 ( 0 ) = 0110 µ 3 ( 0 ) = 01101001 µ 4 ( 0 ) = 0110100110010110 . . . µ ω ( 0 ) = 0110100110010110 · · ·
T HE T HUE -M ORSE WORD µ ω ( 0 ) = 0110100110010110 · · ·
T HE T HUE -M ORSE WORD µ ω ( 0 ) = 0110100110010110 · · · ◮ This word contains no cubes.
T HE T HUE -M ORSE WORD µ ω ( 0 ) = 0110100110010110 · · · ◮ This word contains no cubes. ◮ In fact, it contains no fractional powers larger than 2.
T HE T HUE -M ORSE WORD µ ω ( 0 ) = 0110100110010110 · · · ◮ This word contains no cubes. ◮ In fact, it contains no fractional powers larger than 2. ◮ Example: alfalfa is a 7 / 3-power
T HE T HUE -M ORSE WORD µ ω ( 0 ) = 0110100110010110 · · · ◮ This word contains no cubes. ◮ In fact, it contains no fractional powers larger than 2. ◮ Example: alfalfa is a 7 / 3-power
T HE T HUE -M ORSE WORD µ ω ( 0 ) = 0110100110010110 · · · ◮ This word contains no cubes. ◮ In fact, it contains no fractional powers larger than 2. ◮ Example: alfalfa is a 7 / 3-power
T HE T HUE -M ORSE WORD µ ω ( 0 ) = 0110100110010110 · · · ◮ This word contains no cubes. ◮ In fact, it contains no fractional powers larger than 2. ◮ Example: alfalfa is a 7 / 3-power
T HE T HUE -M ORSE WORD µ ω ( 0 ) = 0110100110010110 · · · ◮ This word contains no cubes. ◮ In fact, it contains no fractional powers larger than 2. ◮ Example: alfalfa is a 7 / 3-power 7 / 3 =
T HE T HUE -M ORSE WORD µ ω ( 0 ) = 0110100110010110 · · · ◮ This word contains no cubes. ◮ In fact, it contains no fractional powers larger than 2. ◮ Example: alfalfa is a 7 / 3-power 7 / 3 = ◮ Pop quiz: 01010 is a...
T HE T HUE -M ORSE WORD µ ω ( 0 ) = 0110100110010110 · · · ◮ This word contains no cubes. ◮ In fact, it contains no fractional powers larger than 2. ◮ Example: alfalfa is a 7 / 3-power 7 / 3 = ◮ Pop quiz: 01010 is a... 5 / 2-power.
T HE T HUE -M ORSE WORD µ ω ( 0 ) = 0110100110010110 · · · ◮ This word contains no cubes. ◮ In fact, it contains no fractional powers larger than 2. ◮ Example: alfalfa is a 7 / 3-power 7 / 3 = ◮ Pop quiz: 01010 is a... 5 / 2-power. ◮ Notice: The Thue-Morse word has many squares, but every square is followed by a letter that breaks the repetition.
T HE T HUE -M ORSE WORD µ ω ( 0 ) = 0110100110010110 · · · ◮ This word contains no cubes. ◮ In fact, it contains no fractional powers larger than 2. ◮ Example: alfalfa is a 7 / 3-power 7 / 3 = ◮ Pop quiz: 01010 is a... 5 / 2-power. ◮ Notice: The Thue-Morse word has many squares, but every square is followed by a letter that breaks the repetition.
T HE T HUE -M ORSE WORD µ ω ( 0 ) = 0110100110010110 · · · ◮ This word contains no cubes. ◮ In fact, it contains no fractional powers larger than 2. ◮ Example: alfalfa is a 7 / 3-power 7 / 3 = ◮ Pop quiz: 01010 is a... 5 / 2-power. ◮ Notice: The Thue-Morse word has many squares, but every square is followed by a letter that breaks the repetition.
T HE T HUE -M ORSE WORD µ ω ( 0 ) = 0110100110010110 · · · ◮ This word contains no cubes. ◮ In fact, it contains no fractional powers larger than 2. ◮ Example: alfalfa is a 7 / 3-power 7 / 3 = ◮ Pop quiz: 01010 is a... 5 / 2-power. ◮ Notice: The Thue-Morse word has many squares, but every square is followed by a letter that breaks the repetition.
T HE T HUE -M ORSE WORD µ ω ( 0 ) = 0110100110010110 · · · ◮ This word contains no cubes. ◮ In fact, it contains no fractional powers larger than 2. ◮ Example: alfalfa is a 7 / 3-power 7 / 3 = ◮ Pop quiz: 01010 is a... 5 / 2-power. ◮ Notice: The Thue-Morse word has many squares, but every square is followed by a letter that breaks the repetition.
T HE T HUE -M ORSE WORD µ ω ( 0 ) = 0110100110010110 · · · ◮ This word contains no cubes. ◮ In fact, it contains no fractional powers larger than 2. ◮ Example: alfalfa is a 7 / 3-power 7 / 3 = ◮ Pop quiz: 01010 is a... 5 / 2-power. ◮ Notice: The Thue-Morse word has many squares, but every square is followed by a letter that breaks the repetition.
T HE T HUE -M ORSE WORD µ ω ( 0 ) = 0110100110010110 · · · ◮ This word contains no cubes. ◮ In fact, it contains no fractional powers larger than 2. ◮ Example: alfalfa is a 7 / 3-power 7 / 3 = ◮ Pop quiz: 01010 is a... 5 / 2-power. ◮ Notice: The Thue-Morse word has many squares, but every square is followed by a letter that breaks the repetition.
C RITICAL EXPONENTS AND REPETITION THRESHOLDS
C RITICAL EXPONENTS AND REPETITION THRESHOLDS ◮ The critical exponent of a word w is defined as sup { r ∈ Q : w contains an r -power } .
C RITICAL EXPONENTS AND REPETITION THRESHOLDS ◮ The critical exponent of a word w is defined as sup { r ∈ Q : w contains an r -power } . ◮ e.g., the critical exponent of the Thue-Morse word is 2.
C RITICAL EXPONENTS AND REPETITION THRESHOLDS ◮ The critical exponent of a word w is defined as sup { r ∈ Q : w contains an r -power } . ◮ e.g., the critical exponent of the Thue-Morse word is 2. ◮ The repetition threshold for a set of words L is the smallest critical exponent among all infinite words in L .
C RITICAL EXPONENTS AND REPETITION THRESHOLDS ◮ The critical exponent of a word w is defined as sup { r ∈ Q : w contains an r -power } . ◮ e.g., the critical exponent of the Thue-Morse word is 2. ◮ The repetition threshold for a set of words L is the smallest critical exponent among all infinite words in L . ◮ e.g., the repetition threshold for the set of all binary words is 2.
A STRUCTURE THEOREM ◮ Question: Are there other infinite binary words with critical exponent 2? What do they look like?
A STRUCTURE THEOREM ◮ Question: Are there other infinite binary words with critical exponent 2? What do they look like? ◮ Answer: It turns out that every infinite binary word with critical exponent less than 7 / 3 looks almost like the Thue-Morse word!
A STRUCTURE THEOREM ◮ Question: Are there other infinite binary words with critical exponent 2? What do they look like? ◮ Answer: It turns out that every infinite binary word with critical exponent less than 7 / 3 looks almost like the Thue-Morse word! Theorem (Karhum¨ aki and Shallit, 2004): Let w be an infinite binary word with critical exponent less than 7 / 3. For every n ≥ 1, a suffix of w has the form µ n ( w n ) for some infinite binary word w n .
A QUICK REVIEW
A QUICK REVIEW ◮ Every long enough binary word contains a square.
A QUICK REVIEW ◮ Every long enough binary word contains a square. ◮ The Thue-Morse word contains nothing “bigger” than a square; it has critical exponent 2.
A QUICK REVIEW ◮ Every long enough binary word contains a square. ◮ The Thue-Morse word contains nothing “bigger” than a square; it has critical exponent 2. ◮ This means that the repetition threshold for the set of all binary words is 2.
A QUICK REVIEW ◮ Every long enough binary word contains a square. ◮ The Thue-Morse word contains nothing “bigger” than a square; it has critical exponent 2. ◮ This means that the repetition threshold for the set of all binary words is 2. ◮ If an infinite binary word has critical exponent less than 7 / 3, then it looks like the Thue-Morse word.
P LAN W ORDS AND R EPETITIONS R ICH WORDS
R ICH WORDS
R ICH WORDS ◮ A palindrome is a finite word that reads the same forwards and backwards.
R ICH WORDS ◮ A palindrome is a finite word that reads the same forwards and backwards. ◮ Examples: 1001 ,
R ICH WORDS ◮ A palindrome is a finite word that reads the same forwards and backwards. ◮ Examples: 1001 , 01010 ,
R ICH WORDS ◮ A palindrome is a finite word that reads the same forwards and backwards. ◮ Examples: 1001 , 01010 , kayak ,
R ICH WORDS ◮ A palindrome is a finite word that reads the same forwards and backwards. ◮ Examples: 1001 , 01010 , kayak , racecar
R ICH WORDS ◮ A palindrome is a finite word that reads the same forwards and backwards. ◮ Examples: 1001 , 01010 , kayak , racecar Theorem (Droubay, Justin, Pirillo 2001): Every word of length n contains at most n distinct nonempty palindromes as factors.
R ICH WORDS ◮ A palindrome is a finite word that reads the same forwards and backwards. ◮ Examples: 1001 , 01010 , kayak , racecar Theorem (Droubay, Justin, Pirillo 2001): Every word of length n contains at most n distinct nonempty palindromes as factors. ◮ A finite word of length n is called rich if it contains n distinct nonempty palindromes.
R ICH WORDS ◮ A palindrome is a finite word that reads the same forwards and backwards. ◮ Examples: 1001 , 01010 , kayak , racecar Theorem (Droubay, Justin, Pirillo 2001): Every word of length n contains at most n distinct nonempty palindromes as factors. ◮ A finite word of length n is called rich if it contains n distinct nonempty palindromes. ◮ The word 01101 contains the palindromes
R ICH WORDS ◮ A palindrome is a finite word that reads the same forwards and backwards. ◮ Examples: 1001 , 01010 , kayak , racecar Theorem (Droubay, Justin, Pirillo 2001): Every word of length n contains at most n distinct nonempty palindromes as factors. ◮ A finite word of length n is called rich if it contains n distinct nonempty palindromes. ◮ The word 01101 contains the palindromes 0 ,
R ICH WORDS ◮ A palindrome is a finite word that reads the same forwards and backwards. ◮ Examples: 1001 , 01010 , kayak , racecar Theorem (Droubay, Justin, Pirillo 2001): Every word of length n contains at most n distinct nonempty palindromes as factors. ◮ A finite word of length n is called rich if it contains n distinct nonempty palindromes. ◮ The word 01101 contains the palindromes 0 , 1 ,
R ICH WORDS ◮ A palindrome is a finite word that reads the same forwards and backwards. ◮ Examples: 1001 , 01010 , kayak , racecar Theorem (Droubay, Justin, Pirillo 2001): Every word of length n contains at most n distinct nonempty palindromes as factors. ◮ A finite word of length n is called rich if it contains n distinct nonempty palindromes. ◮ The word 01101 contains the palindromes 0 , 1 , 11 ,
R ICH WORDS ◮ A palindrome is a finite word that reads the same forwards and backwards. ◮ Examples: 1001 , 01010 , kayak , racecar Theorem (Droubay, Justin, Pirillo 2001): Every word of length n contains at most n distinct nonempty palindromes as factors. ◮ A finite word of length n is called rich if it contains n distinct nonempty palindromes. ◮ The word 01101 contains the palindromes 0 , 1 , 11 , 0110 ,
R ICH WORDS ◮ A palindrome is a finite word that reads the same forwards and backwards. ◮ Examples: 1001 , 01010 , kayak , racecar Theorem (Droubay, Justin, Pirillo 2001): Every word of length n contains at most n distinct nonempty palindromes as factors. ◮ A finite word of length n is called rich if it contains n distinct nonempty palindromes. ◮ The word 01101 contains the palindromes 0 , 1 , 11 , 0110 , and 101 ,
R ICH WORDS ◮ A palindrome is a finite word that reads the same forwards and backwards. ◮ Examples: 1001 , 01010 , kayak , racecar Theorem (Droubay, Justin, Pirillo 2001): Every word of length n contains at most n distinct nonempty palindromes as factors. ◮ A finite word of length n is called rich if it contains n distinct nonempty palindromes. ◮ The word 01101 contains the palindromes 0 , 1 , 11 , 0110 , and 101 , so it is rich.
R ICH WORDS ◮ A palindrome is a finite word that reads the same forwards and backwards. ◮ Examples: 1001 , 01010 , kayak , racecar Theorem (Droubay, Justin, Pirillo 2001): Every word of length n contains at most n distinct nonempty palindromes as factors. ◮ A finite word of length n is called rich if it contains n distinct nonempty palindromes. ◮ The word 01101 contains the palindromes 0 , 1 , 11 , 0110 , and 101 , so it is rich. ◮ The word 0120 contains only the palindromes 0 , 1 , and 2 , so it is not rich.
R ICH WORDS ◮ A palindrome is a finite word that reads the same forwards and backwards. ◮ Examples: 1001 , 01010 , kayak , racecar Theorem (Droubay, Justin, Pirillo 2001): Every word of length n contains at most n distinct nonempty palindromes as factors. ◮ A finite word of length n is called rich if it contains n distinct nonempty palindromes. ◮ The word 01101 contains the palindromes 0 , 1 , 11 , 0110 , and 101 , so it is rich. ◮ The word 0120 contains only the palindromes 0 , 1 , and 2 , so it is not rich. ◮ An infinite word is called rich if all of its finite factors are rich.
R EPETITIONS IN RICH WORDS Theorem (Pelantov´ a and Starosta, 2013): Every infinite rich word contains a square.
R EPETITIONS IN RICH WORDS Theorem (Pelantov´ a and Starosta, 2013): Every infinite rich word contains a square. ◮ This result holds over any finite alphabet.
R EPETITIONS IN RICH WORDS Theorem (Pelantov´ a and Starosta, 2013): Every infinite rich word contains a square. ◮ This result holds over any finite alphabet. ◮ So, what types of powers can be avoided by infinite rich words on k letters?
R EPETITIONS IN RICH WORDS Theorem (Pelantov´ a and Starosta, 2013): Every infinite rich word contains a square. ◮ This result holds over any finite alphabet. ◮ So, what types of powers can be avoided by infinite rich words on k letters? ◮ Cubes?
R EPETITIONS IN RICH WORDS Theorem (Pelantov´ a and Starosta, 2013): Every infinite rich word contains a square. ◮ This result holds over any finite alphabet. ◮ So, what types of powers can be avoided by infinite rich words on k letters? ◮ Cubes? ◮ If so, what about fractional powers between 2 and 3?
R EPETITIONS IN RICH WORDS Theorem (Pelantov´ a and Starosta, 2013): Every infinite rich word contains a square. ◮ This result holds over any finite alphabet. ◮ So, what types of powers can be avoided by infinite rich words on k letters? ◮ Cubes? ◮ If so, what about fractional powers between 2 and 3? ◮ We are asking for the repetition threshold for rich words on k letters, denoted RRT ( k ) .
R EPETITIONS IN RICH WORDS Theorem (Pelantov´ a and Starosta, 2013): Every infinite rich word contains a square. ◮ This result holds over any finite alphabet. ◮ So, what types of powers can be avoided by infinite rich words on k letters? ◮ Cubes? ◮ If so, what about fractional powers between 2 and 3? ◮ We are asking for the repetition threshold for rich words on k letters, denoted RRT ( k ) . ◮ We will determine RRT ( 2 ) .
R EPETITIONS IN RICH WORDS Theorem (Baranwal and Shallit, 2019): There is an infinite √ binary rich word with critical exponent 2 + 2 / 2.
R EPETITIONS IN RICH WORDS Theorem (Baranwal and Shallit, 2019): There is an infinite √ binary rich word with critical exponent 2 + 2 / 2. √ ◮ Note: 2 + 2 / 2 ≈ 2 . 707.
R EPETITIONS IN RICH WORDS Theorem (Baranwal and Shallit, 2019): There is an infinite √ binary rich word with critical exponent 2 + 2 / 2. √ ◮ Note: 2 + 2 / 2 ≈ 2 . 707. ◮ They conjectured that this is the smallest possible critical exponent among infinite binary rich words, i.e., that √ RRT ( 2 ) = 2 + 2 / 2.
R EPETITIONS IN RICH WORDS Theorem (Baranwal and Shallit, 2019): There is an infinite √ binary rich word with critical exponent 2 + 2 / 2. √ ◮ Note: 2 + 2 / 2 ≈ 2 . 707. ◮ They conjectured that this is the smallest possible critical exponent among infinite binary rich words, i.e., that √ RRT ( 2 ) = 2 + 2 / 2. √ ◮ The irrationality of 2 + 2 / 2 makes this hard to prove!
R EPETITIONS IN RICH WORDS Theorem (Baranwal and Shallit, 2019): There is an infinite √ binary rich word with critical exponent 2 + 2 / 2. √ ◮ Note: 2 + 2 / 2 ≈ 2 . 707. ◮ They conjectured that this is the smallest possible critical exponent among infinite binary rich words, i.e., that √ RRT ( 2 ) = 2 + 2 / 2. √ ◮ The irrationality of 2 + 2 / 2 makes this hard to prove! ◮ Baranwal and Shallit: RRT ( 2 ) ≥ 2 . 7
B ARANWAL AND S HALLIT ’ S CONSTRUCTION Define morphisms f and h by f ( 0 ) = 0 f ( 1 ) = 01 f ( 2 ) = 011 h ( 0 ) = 01 h ( 1 ) = 02 h ( 2 ) = 022 .
B ARANWAL AND S HALLIT ’ S CONSTRUCTION Define morphisms f and h by f ( 0 ) = 0 f ( 1 ) = 01 f ( 2 ) = 011 h ( 0 ) = 01 h ( 1 ) = 02 h ( 2 ) = 022 . The infinite word f ( h ω ( 0 )) is rich and has critical exponent √ 2 + 2 / 2.
B ARANWAL AND S HALLIT ’ S CONSTRUCTION Define morphisms f and h by f ( 0 ) = 0 f ( 1 ) = 01 f ( 2 ) = 011 h ( 0 ) = 01 h ( 1 ) = 02 h ( 2 ) = 022 . The infinite word f ( h ω ( 0 )) is rich and has critical exponent √ 2 + 2 / 2. ◮ The proof was completed using the automatic theorem proving software Walnut .
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