Martine Quefflec University of Lille France For Christian - PowerPoint PPT Presentation

Martine Queffélec University of Lille – France For Christian – 7/11/2019 M. Queffélec (Lille 1) For Christian – 7/11/2019 1 / 16

Normal sets Let Λ = ( t n ) be a sequence of positive integers (more generally real numbers). M. Queffélec (Lille 1) For Christian – 7/11/2019 2 / 16

Normal sets Let Λ = ( t n ) be a sequence of positive integers (more generally real numbers). Definition The normal set associated to Λ is B (Λ) = { x ∈ R , ( t n x ) u.d. mod 1 } equivalently (Weyl’s criterion) � B (Λ) = { x ∈ R , ∀ k � = 0 , 1 e ( kt n x ) → 0 . } N n ≤ N M. Queffélec (Lille 1) For Christian – 7/11/2019 2 / 16

Normal sets Let Λ = ( t n ) be a sequence of positive integers (more generally real numbers). Definition The normal set associated to Λ is B (Λ) = { x ∈ R , ( t n x ) u.d. mod 1 } equivalently (Weyl’s criterion) � B (Λ) = { x ∈ R , ∀ k � = 0 , 1 e ( kt n x ) → 0 . } N n ≤ N First examples. B ( N ) = R \ Q ; the same holds for Λ = ( P ( n )) with P ∈ Z [ X ]. 1 M. Queffélec (Lille 1) For Christian – 7/11/2019 2 / 16

Normal sets Let Λ = ( t n ) be a sequence of positive integers (more generally real numbers). Definition The normal set associated to Λ is B (Λ) = { x ∈ R , ( t n x ) u.d. mod 1 } equivalently (Weyl’s criterion) � B (Λ) = { x ∈ R , ∀ k � = 0 , 1 e ( kt n x ) → 0 . } N n ≤ N First examples. B ( N ) = R \ Q ; the same holds for Λ = ( P ( n )) with P ∈ Z [ X ]. 1 If q ≥ 2 and t n = q n , then B (Λ) =: N q ( normal numbers to base q ) with negligible 2 – but uncountable – complement set (Borel). M. Queffélec (Lille 1) For Christian – 7/11/2019 2 / 16

Normal sets Let Λ = ( t n ) be a sequence of positive integers (more generally real numbers). Definition The normal set associated to Λ is B (Λ) = { x ∈ R , ( t n x ) u.d. mod 1 } equivalently (Weyl’s criterion) � B (Λ) = { x ∈ R , ∀ k � = 0 , 1 e ( kt n x ) → 0 . } N n ≤ N First examples. B ( N ) = R \ Q ; the same holds for Λ = ( P ( n )) with P ∈ Z [ X ]. 1 If q ≥ 2 and t n = q n , then B (Λ) =: N q ( normal numbers to base q ) with negligible 2 – but uncountable – complement set (Borel). For intermediate growth rate ? For example, the Furstenberg sequence : 3 ( s n ) = { 2 j 3 k , j ≥ 1 , k ≥ 1 } re-arranged in increasing order? M. Queffélec (Lille 1) For Christian – 7/11/2019 2 / 16

Let B ⊂ R . M. Queffélec (Lille 1) For Christian – 7/11/2019 3 / 16

Let B ⊂ R . Definition (MMF) B is a normal set if there exists Λ such that B = B (Λ) . M. Queffélec (Lille 1) For Christian – 7/11/2019 3 / 16

Let B ⊂ R . Definition (MMF) B is a normal set if there exists Λ such that B = B (Λ) . What does a normal set look like ? M. Queffélec (Lille 1) For Christian – 7/11/2019 3 / 16

Let B ⊂ R . Definition (MMF) B is a normal set if there exists Λ such that B = B (Λ) . What does a normal set look like ? G. Rauzy (1970) gives the following description : M. Queffélec (Lille 1) For Christian – 7/11/2019 3 / 16

Let B ⊂ R . Definition (MMF) B is a normal set if there exists Λ such that B = B (Λ) . What does a normal set look like ? G. Rauzy (1970) gives the following description : Theorem B ⊂ R is a normal set if and only if ∈ B , B + 1 = B. 0 / 1 M. Queffélec (Lille 1) For Christian – 7/11/2019 3 / 16

Let B ⊂ R . Definition (MMF) B is a normal set if there exists Λ such that B = B (Λ) . What does a normal set look like ? G. Rauzy (1970) gives the following description : Theorem B ⊂ R is a normal set if and only if ∈ B , B + 1 = B. 0 / 1 ∀ q ∈ Z ∗ , qB ⊂ B. 2 M. Queffélec (Lille 1) For Christian – 7/11/2019 3 / 16

Let B ⊂ R . Definition (MMF) B is a normal set if there exists Λ such that B = B (Λ) . What does a normal set look like ? G. Rauzy (1970) gives the following description : Theorem B ⊂ R is a normal set if and only if ∈ B , B + 1 = B. 0 / 1 ∀ q ∈ Z ∗ , qB ⊂ B. 2 There exists a sequence of continuous functions on R , ( f n ) , such that 3 n →∞ f n ( x ) = 0 ⇐ lim ⇒ x ∈ B . M. Queffélec (Lille 1) For Christian – 7/11/2019 3 / 16

Let B ⊂ R . Definition (MMF) B is a normal set if there exists Λ such that B = B (Λ) . What does a normal set look like ? G. Rauzy (1970) gives the following description : Theorem B ⊂ R is a normal set if and only if ∈ B , B + 1 = B. 0 / 1 ∀ q ∈ Z ∗ , qB ⊂ B. 2 There exists a sequence of continuous functions on R , ( f n ) , such that 3 n →∞ f n ( x ) = 0 ⇐ lim ⇒ x ∈ B . M. Queffélec (Lille 1) For Christian – 7/11/2019 3 / 16

Let B ⊂ R . Definition (MMF) B is a normal set if there exists Λ such that B = B (Λ) . What does a normal set look like ? G. Rauzy (1970) gives the following description : Theorem B ⊂ R is a normal set if and only if ∈ B , B + 1 = B. 0 / 1 ∀ q ∈ Z ∗ , qB ⊂ B. 2 There exists a sequence of continuous functions on R , ( f n ) , such that 3 n →∞ f n ( x ) = 0 ⇐ lim ⇒ x ∈ B . Comments. 1. No information on the associated sequence Λ. 2. No such result if we impose Λ to be an increasing sequence. M. Queffélec (Lille 1) For Christian – 7/11/2019 3 / 16

Christian’s result on substitutive normal sets M. Queffélec (Lille 1) For Christian – 7/11/2019 4 / 16

Christian’s result on substitutive normal sets In 1980, together with Christol, Kamae and Mendes France, G. Rauzy has written a founding paper on algebraic numbers, automata and substitutions. M. Queffélec (Lille 1) For Christian – 7/11/2019 4 / 16

Christian’s result on substitutive normal sets In 1980, together with Christol, Kamae and Mendes France, G. Rauzy has written a founding paper on algebraic numbers, automata and substitutions. Unsurprinsingly, Rauzy proposed to his student, Christian, the study of sequences of integers defined by some automaton or substitution. In particular, can we describe the associated normal sets ? M. Queffélec (Lille 1) For Christian – 7/11/2019 4 / 16

Christian’s result on substitutive normal sets In 1980, together with Christol, Kamae and Mendes France, G. Rauzy has written a founding paper on algebraic numbers, automata and substitutions. Unsurprinsingly, Rauzy proposed to his student, Christian, the study of sequences of integers defined by some automaton or substitution. In particular, can we describe the associated normal sets ? In 1989 Christian brings a complete answer to this problem in the irreducible case. M. Queffélec (Lille 1) For Christian – 7/11/2019 4 / 16

Christian’s result on substitutive normal sets In 1980, together with Christol, Kamae and Mendes France, G. Rauzy has written a founding paper on algebraic numbers, automata and substitutions. Unsurprinsingly, Rauzy proposed to his student, Christian, the study of sequences of integers defined by some automaton or substitution. In particular, can we describe the associated normal sets ? In 1989 Christian brings a complete answer to this problem in the irreducible case. Theorem B ⊂ R is a normal set associated to some irreducible substitution if and only if R \ B is a finite real field extension of Q . M. Queffélec (Lille 1) For Christian – 7/11/2019 4 / 16

Christian’s result on substitutive normal sets In 1980, together with Christol, Kamae and Mendes France, G. Rauzy has written a founding paper on algebraic numbers, automata and substitutions. Unsurprinsingly, Rauzy proposed to his student, Christian, the study of sequences of integers defined by some automaton or substitution. In particular, can we describe the associated normal sets ? In 1989 Christian brings a complete answer to this problem in the irreducible case. Theorem B ⊂ R is a normal set associated to some irreducible substitution if and only if R \ B is a finite real field extension of Q . We need some clarification... M. Queffélec (Lille 1) For Christian – 7/11/2019 4 / 16

Reminds on substitutions Let ζ be a substitution on a finite alphabet A of cardinality s (a map A → A ∗ extended by concatenation). A is identified to { 1 , 2 , . . . , s } . M := M ( ζ ) is the s × s -matrix with entries 1 m i , j = # { j ∈ ζ ( i ) } , i , j ∈ A . M. Queffélec (Lille 1) For Christian – 7/11/2019 5 / 16

Reminds on substitutions Let ζ be a substitution on a finite alphabet A of cardinality s (a map A → A ∗ extended by concatenation). A is identified to { 1 , 2 , . . . , s } . M := M ( ζ ) is the s × s -matrix with entries 1 m i , j = # { j ∈ ζ ( i ) } , i , j ∈ A . M (or ζ ) is irreducible if, for every ( i , j ) there exists n = n ( i , j ) such that j appears 2 in ζ n ( i ) i.e. ( M n ) i , j > 0. M. Queffélec (Lille 1) For Christian – 7/11/2019 5 / 16

Reminds on substitutions Let ζ be a substitution on a finite alphabet A of cardinality s (a map A → A ∗ extended by concatenation). A is identified to { 1 , 2 , . . . , s } . M := M ( ζ ) is the s × s -matrix with entries 1 m i , j = # { j ∈ ζ ( i ) } , i , j ∈ A . M (or ζ ) is irreducible if, for every ( i , j ) there exists n = n ( i , j ) such that j appears 2 in ζ n ( i ) i.e. ( M n ) i , j > 0. i , j > 0 for every ( i , j ) (or M n > 0). M (or ζ ) is primitive if n exists such that M n 3 M. Queffélec (Lille 1) For Christian – 7/11/2019 5 / 16