Balance properties of infinite words associated with quadratic Pisot numbers Ondˇ rej Turek Faculty of Nuclear Sciences and Physical Engineering Czech Technical University in Prague Journ´ ees Num´ eration 2008 Ondˇ rej Turek Balance properties of infinite words associated with quadratic Pisot
Plan of the talk Balance properties of infinite words associated with quadratic Pisot numbers 1 Simple and non-simple quadratic Pisot numbers 2 The set of β -integers, infinite word associated to a quadratic Pisot number 3 Balance properties: k -balanced word 4 Theorem 5 Sketch of the proof Ondˇ rej Turek Balance properties of infinite words associated with quadratic Pisot
Quadratic Pisot numbers β is a quadratic Pisot number iff β is algebraic integer greater than 1 Ondˇ rej Turek Balance properties of infinite words associated with quadratic Pisot
Quadratic Pisot numbers β is a quadratic Pisot number iff β is algebraic integer greater than 1 of degree 2 Ondˇ rej Turek Balance properties of infinite words associated with quadratic Pisot
Quadratic Pisot numbers β is a quadratic Pisot number iff β is algebraic integer greater than 1 of degree 2 such that its Galois conjugate has modulus less than 1. Ondˇ rej Turek Balance properties of infinite words associated with quadratic Pisot
Quadratic Pisot numbers β is a quadratic Pisot number iff β is algebraic integer greater than 1 of degree 2 such that its Galois conjugate has modulus less than 1. R´ enyi β -expansion of unity is the lexicographically greatest sequence d β (1) = t 1 t 2 t 3 · · · , where t i ∈ N such that 1 = � i ≥ 1 t i β − i . Ondˇ rej Turek Balance properties of infinite words associated with quadratic Pisot
Quadratic Pisot numbers β is a quadratic Pisot number iff β is algebraic integer greater than 1 of degree 2 such that its Galois conjugate has modulus less than 1. R´ enyi β -expansion of unity is the lexicographically greatest sequence d β (1) = t 1 t 2 t 3 · · · , where t i ∈ N such that 1 = � i ≥ 1 t i β − i . For β being quadratic Pisot, d β (1) can be finite , d β (1) = pq , p ≥ q ≥ 1 - occurs when β is a simple quadratic Pisot number, Ondˇ rej Turek Balance properties of infinite words associated with quadratic Pisot
Quadratic Pisot numbers β is a quadratic Pisot number iff β is algebraic integer greater than 1 of degree 2 such that its Galois conjugate has modulus less than 1. R´ enyi β -expansion of unity is the lexicographically greatest sequence d β (1) = t 1 t 2 t 3 · · · , where t i ∈ N such that 1 = � i ≥ 1 t i β − i . For β being quadratic Pisot, d β (1) can be finite , d β (1) = pq , p ≥ q ≥ 1 - occurs when β is a simple quadratic Pisot number, eventually periodic , d β (1) = pq ω , p > q ≥ 1 (= pq ω denotes pqqq · · · ) (no other possibility) Ondˇ rej Turek Balance properties of infinite words associated with quadratic Pisot
Quadratic Pisot numbers β is a quadratic Pisot number iff β is algebraic integer greater than 1 of degree 2 such that its conjugate element has modulus less than 1. R´ enyi β -expansion of unity is the lexicographically greatest sequence d β (1) = t 1 t 2 t 3 · · · , where t i ∈ N such that 1 = � i ≥ 1 t i β − i . For β being quadratic Pisot, d β (1) can be finite , d β (1) = pq , p ≥ q ≥ 1 ◮ β is called a simple quadratic Pisot number, eventually periodic , d β (1) = pq ω , p > q ≥ 1 Ondˇ rej Turek Balance properties of infinite words associated with quadratic Pisot
Quadratic Pisot numbers β is a quadratic Pisot number iff β is algebraic integer greater than 1 of degree 2 such that its conjugate element has modulus less than 1. R´ enyi β -expansion of unity is the lexicographically greatest sequence d β (1) = t 1 t 2 t 3 · · · , where t i ∈ N such that 1 = � i ≥ 1 t i β − i . For β being quadratic Pisot, d β (1) can be finite , d β (1) = pq , p ≥ q ≥ 1 ◮ β is called a simple quadratic Pisot number, eventually periodic , d β (1) = pq ω , p > q ≥ 1 ◮ β is called a non-simple quadratic Pisot number. (no other possibility) Ondˇ rej Turek Balance properties of infinite words associated with quadratic Pisot
β -integers A β -expansion of x ≥ 0 is a representation of the form x = x k β k + x k − 1 β k − 1 + · · · + x 0 + x − 1 β − 1 + x − 2 β − 2 + · · · , where x i ∈ N 0 are obtained by the ‘greedy algorithm’ . Ondˇ rej Turek Balance properties of infinite words associated with quadratic Pisot
β -integers A β -expansion of x ≥ 0 is a representation of the form x = x k β k + x k − 1 β k − 1 + · · · + x 0 + x − 1 β − 1 + x − 2 β − 2 + · · · , where x i ∈ N 0 are obtained by the ‘greedy algorithm’ . We denote: � x � β = x k x k − 1 · · · x 0 • x − 1 x − 2 · · · The set of non-negative β -integers : � Z + � � x � β = x k x k − 1 · · · x 0 •} β = { x ≥ 0 Ondˇ rej Turek Balance properties of infinite words associated with quadratic Pisot
β -integers A β -expansion of x ≥ 0 is a representation of the form x = x k β k + x k − 1 β k − 1 + · · · + x 0 + x − 1 β − 1 + x − 2 β − 2 + · · · , where x i ∈ N 0 are obtained by the ‘greedy algorithm’ . We denote: � x � β = x k x k − 1 · · · x 0 • x − 1 x − 2 · · · The set of non-negative β -integers : � Z + � � x � β = x k x k − 1 · · · x 0 •} β = { x ≥ 0 Theorem: There are exactly two types of distances between neighboring points of Z + β on the real line, namely ∆ A = 1, ∆ B = β − ⌊ β ⌋ . Ondˇ rej Turek Balance properties of infinite words associated with quadratic Pisot
Generating substitutions for the word u β Let us assign letters A and B to ∆ A , ∆ B : the order of distances in Z + β defines an infinite word u β Ondˇ rej Turek Balance properties of infinite words associated with quadratic Pisot
Generating substitutions for the word u β Let us assign letters A and B to ∆ A , ∆ B : the order of distances in Z + β defines an infinite word u β If β is simple , then u β is a fixed point of ϕ ( A ) = A p B , ϕ ( B ) = A q , p ≥ q ≥ 1 . A �→ A p B �→ ( A p B ) p A q �→ · · · ( A p denotes A · · · A ) � �� � p Ondˇ rej Turek Balance properties of infinite words associated with quadratic Pisot
Generating substitutions for the word u β Let us assign letters A and B to ∆ A , ∆ B : the order of distances in Z + β defines an infinite word u β If β is simple , then u β is a fixed point of ϕ ( A ) = A p B , ϕ ( B ) = A q , p ≥ q ≥ 1 . A �→ A p B �→ ( A p B ) p A q �→ · · · ( A p denotes A · · · A ) � �� � p If β is non-simple , then u β is a fixed point of ϕ ( A ) = A p B , ϕ ( B ) = A q B , p > q ≥ 1 . A �→ A p B �→ ( A p B ) p A q B �→ · · · Ondˇ rej Turek Balance properties of infinite words associated with quadratic Pisot
k -balanced word Notation: length of a word w = w 1 w 2 · · · w n : | w | = n the number of letters A in the word w : | w | A factor of the word v (finite or infinite): v = w (1) ww (2) prefix of v : v = ww (2) suffix of v : v = w (1) w Ondˇ rej Turek Balance properties of infinite words associated with quadratic Pisot
k -balanced word Notation: length of a word w = w 1 w 2 · · · w n : | w | = n the number of letters A in the word w : | w | A factor of the word v (finite or infinite): v = w (1) ww (2) prefix of v : v = ww (2) suffix of v : v = w (1) w Definition: A word u in the binary alphabet A = { A , B } is k -balanced, if for every pair of factors w , ˆ w of u , it holds | w | = | ˆ w | ⇒ | | w | A − | ˆ w | A | ≤ k . Ondˇ rej Turek Balance properties of infinite words associated with quadratic Pisot
k -balanced word Notation: length of a word w = w 1 w 2 · · · w n : | w | = n the number of letters A in the word w : | w | A factor of the word v (finite or infinite): v = w (1) ww (2) prefix of v : v = ww (2) suffix of v : v = w (1) w Definition: A word u in the binary alphabet A = { A , B } is k -balanced, if for every pair of factors w , ˆ w of u , it holds | w | = | ˆ w | ⇒ | | w | A − | ˆ w | A | ≤ k . Questions: 1 Are the words u β k -balanced for some k ? 2 If yes, what is the minimal k ? Ondˇ rej Turek Balance properties of infinite words associated with quadratic Pisot
Balance properties of u β Theorem (Adamczewski): For every quadratic Pisot number β there is a k ∈ N such that u β is k -balanced. Ondˇ rej Turek Balance properties of infinite words associated with quadratic Pisot
Balance properties of u β Theorem (Adamczewski): For every quadratic Pisot number β there is a k ∈ N such that u β is k -balanced. Theorem (our result) If β is a quadratic simple Pisot number, d β (1) = pq, then � � �� p − 1 1 + u β is -balanced . p + 1 − q Ondˇ rej Turek Balance properties of infinite words associated with quadratic Pisot
Balance properties of u β Theorem (Adamczewski): For every quadratic Pisot number β there is a k ∈ N such that u β is k -balanced. Theorem (our result) If β is a quadratic simple Pisot number, d β (1) = pq, then � � �� p − 1 1 + u β is -balanced . p + 1 − q If β is a quadratic non-simple Pisot number, d β (1) = pq ω , then �� p − 1 �� u β is -balanced . q Ondˇ rej Turek Balance properties of infinite words associated with quadratic Pisot
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