Automatic sequences, generalised polynomials & nilmanifolds - PowerPoint PPT Presentation

Automatic sequences, generalised polynomials & nilmanifolds Jakub Konieczny Univeristy of Oxford Jagiellonian University Hebrew University of Jerusalem 8 Kongres Polskiego Towarzystwa Matematycznego Lublin, 22 September 2017

Motivation Theorem (Kronecker) Let θ ∈ R d . Then, the sequence { nθ } = ( { nθ 1 } , . . . , { nθ d } ) for n ∈ N is equidistributed in [0 , 1) d unless there exists k ∈ Z d \ { 0 } such that k · θ ∈ Z . Notation: { x } = x − ⌊ x ⌋ for x ∈ R ; { x } = ( { x 1 } , . . . , { x d } ) for x = ( x 1 , . . . , x d ) ∈ R d ; x · y = x 1 y 1 + · · · + x d y d for x, y ∈ R d . A sequence x n ∈ X ( n ∈ N ) is equidistributed with respect to µ ∈ Meas( X ) iff for any f ∈ C ( X ) : 1 � � f ( x n ) → fdµ as N → ∞ . N n<N If X = [0 , 1) d , µ is the Lebesgue measure by default. Remark If k ∈ Z d \ { 0 } and k · θ ∈ Z then k · { nθ } ∈ Z for all n ∈ Z . 2 / 22

Motivation Theorem (Weyl) Let p ( x ) ∈ R [ x ] d . Then, the sequence { p ( n ) } = ( { p 1 ( n ) } , . . . , { p d ( n ) } ) for n ∈ N is equidistributed in [0 , 1) d unless there exists k ∈ Z d \ { 0 } such that k · p ( x ) ∈ Z [ x ] . Remark Identify [0 , 1) d with the d -torus R d / Z d . For k ∈ Z d \ { 0 } , the set x ∈ R d / Z d : k · x ∈ Z � � is a ( d − 1) -torus. A more precise statement is true: If p ( x ) ∈ R [ x ] d then { p ( n ) } is equidistributed in a union of subtorii of R d / Z d . What about more general expressions? Example � √ � √ �� 2 3 n is equidistributed in [0 , 1) . � √ � √ � √ � √ �� �� 2 2 n is not equidistributed in [0 , 1) ; and nor is 2 n 2 n . � √ � √ � 2 � 2 2 n is equidistributed in [0 , 1) . 3 / 22

Motivation Example � √ � √ �� 2 3 n is equidistributed in [0 , 1) . � √ � √ �� √ √ � √ �� √ � √ �� � �� � �� Proof: 2 3 n = 6 n − 2 3 n and 6 n , 3 n is eqdistributed in [0 , 1) 2 . Example � √ � √ � √ � √ �� �� 2 2 n and 2 n 2 n are not equidistributed in [0 , 1) . � √ � √ √ � √ √ � √ � √ � 2 and �� � �� � �� Proof: 2 2 n = − 2 2 n ; 2 2 n 2 n = 2 n � √ � 2 n is eqdistributed in [0 , 1) . Example � √ � √ � 2 � 2 2 n is equidistributed in [0 , 1) . � √ � √ √ √ � √ � 2 � � � 2 � 2 n 2 � � Proof: 2 2 n = − 2 + 2 2 n and √ � √ 2 n 2 � is eqdistributed in [0 , 1) 2 . �� �� 2 , 2 n 4 / 22

Generalised polynomials Definition The generalised polynomial maps Z → R (denoted GP ) are the smallest family such that R [ x ] ⊂ GP and for any g, h ∈ GP : g + h ∈ GP , g · h ∈ GP , ⌊ g ⌋ ∈ GP . (Here: ⌊ g ⌋ ( n ) = ⌊ g ( n ) ⌋ .) If g ∈ GP then also { g } ∈ GP . (Here, { g } ( n ) = { g ( n ) } .) The generalised polynomial maps Z → R d are just d -tuples ( g 1 , . . . , g d ) with g i ∈ GP for 1 ≤ i ≤ d . We call a set E ⊂ N a generalised polynomial set if 1 E ∈ GP . Example � √ 1 2 { n/ 2 } 2 n 2 � is equidistributed with respect to 1 2 δ 0 + 1 2 λ [0 , 1) . 2 ⌊ ( n + 1) θ ⌋ − ⌊ nθ ⌋ is equidistributed with respect to (1 − θ ) δ 0 + θδ 1 . ( 0 < θ < 1 ) 3 � √ � 2 is equidistributed with respect to measure µ with dµ = dt 2 n t . √ 2 5 / 22

Generalised polynomials Question 1 For which g ∈ GP is { g } equidistributed in [0 , 1) ? 2 What are possible distributions of bounded g ∈ GP ? 3 What may a generalised polynomial set look like? Answers: 1 For generic choice of g ∈ GP , { g } is equidistributed in [0 , 1) . There is an algorithmic way to decide ( → Haland-Knutson, Leibman). 2 Distribution of any generalised polynomial is described by an algebraic expression, except for a few exceptional points ( → Bergelson, Leibman). 3 A generalised polynomial set with positive density is always combinatorially rich ( → Bergelson, Leibman). Conversely, a sparse generalised polynomial set is always combinatorially poor. Any sufficiently sparse set is generalised polynomial ( → Byszewski, K.). 6 / 22

Equidistributed generalised polynomials The multiset of coefficients of a generalised polynomial g consists of the constants used to write down g ( x ) . For instance, the coefficients of √ � √ √ � √ � 2 � g 0 ( n ) = 2 3 n + 5 n 7 n √ √ √ √ are 2 , 3 , 5 , 7 . More precisely: If g ( n ) = αn i then coeff( g ) = { α } ; coeff( g + h ) = coeff( g ) ∪ coeff( h ) ; coeff( g ⌊ h ⌋ i ) = coeff( g ) ∪ coeff( h ) . Theorem (Haland–Knutson) Let g ∈ GP be a generalised polynomial with coefficients α 1 , . . . , α r . Suppose that all products � i ∈ I α i for I ⊂ { 1 , 2 , . . . , r } are linearly independent over Q . Then { g ( n ) } is equidistributed in [0 , 1) . For instance, { g 0 ( n ) } is equidistributed because √ √ √ √ √ √ √ √ √ √ √ √ √ √ √ 1 , 2 , 3 , 5 , 7 , 6 , 10 , 14 , 15 , 21 , 35 , 30 , 42 , 70 , 105 , 210 are Q -linearly independent. 7 / 22

Nilmanifolds, nilsequences, and Nil–Bohr sets Definition (nilsequences) Let G be a ( d -step) nilpotent Lie group, and Γ < G a cocompact discrete subgroup. 1 The space X = G/ Γ is a nilmanifold . 2 For g ∈ G , the map T g : X → X , x �→ gx is a nilrotation . 3 The dynamical system ( G/ Γ , T g ) is a nilsystem. It has a natural Haar measure µ G/ Γ which is T g -invariant. 4 If F : X → R is a (smooth) function, x 0 ∈ X , then ψ ( n ) = F ( g n x 0 ) is a ( d -step) nilsequence . A reassuring example : Take G = R , Γ = Z . Then G/ Γ = T , the unit circle, equipped with rotations x �→ x + θ . The additive characters n �→ e ( nθ ) are 1 -step nilsequences. Definition (Nil–Bohr sets) Let ψ be a ( d -step) nilsequence, and V ⊂ R an open set. A set A = { n : ψ ( n ) ∈ V } is called a ( d -step) Nil–Bohr set (if � = ∅ ). If ψ (0) ∈ V (i.e. 0 ∈ A ), then A is called a Nil–Bohr 0 set. 8 / 22

Nilmanifolds and generalised polynomials Theorem (Bergelson–Leibman) Let g : Z → R d be a bounded generalised polynomial. Then, there exists a minimal nilsystem ( G/ Γ , T ) with a point x 0 ∈ G/ Γ and a piecewise polynomial map F : G/ Γ → R d such that g ( n ) = F ( T n x 0 ) . Slogan: bounded GP sequences ≃ nilsequences; GP sets ≃ Nil–Bohr sets. Definition : A semialgebraic set V ⊂ [0 , 1) m is one which can be defined by a finite number of polynomial equalities and inequalities. A map P : [0 , 1) m → R is piecewise polynomial if there exists a partition [0 , 1) m = � i V i into semialgebraic pieces, such that P | V i is a polynomial for each i . A nilmanifold G/ Γ carries a natural system of coordinates ( → Malcev basis), under which it can be identified with [0 , 1) m for some m . Hence, it makes sense to speak of piecewise polynomial map G/ Γ → R d . 9 / 22

Nilmanifolds and generalised polynomials Theorem (Bergelson–Leibman) Let g : Z → R d be a bounded generalised polynomial. Then, there exists a minimal nilsystem ( G/ Γ , T ) with a point x 0 ∈ G/ Γ and a piecewise polynomial map F : G/ Γ → R d such that g ( n ) = F ( T n x 0 ) . Definition : If A ⊂ R m is semialgebraic with non-empty interior and P : A → R d is polynomial, then we call S = P ( A ) a parametrized polynomial set . In this situation S carries a natural measure µ S , the pushforward of the normalised Lebesgue measure on A . A parametrised piecewise polynomial set S ⊂ R d is a finite union of polynomial sets S = � i S i , equipped with measure µ S = � i α i µ S i , α i > 0 . Corollary Let g : Z → R d be a bounded generalised polynomial. Then, there exists a parametrised piecewise polynomial set S ⊂ R d with measure µ S , such that g ( n ) ∈ S for all most all n and g ( n ) is equidistributed in S with respect to µ S . 10 / 22

IP sets Finite sums. For n = ( n i ) ∞ i =1 , n i ∈ N , define: � � � n i : α ⊂ N , finite, α � = ∅ FS( n ) = . i ∈ α A set A ⊂ N is IP is there is n with A ⊃ FS( n ) . A set B ⊂ N is IP ∗ if B ∩ A � = ∅ for any IP set A . Fact 1 Any IP ∗ set is syndetic (i.e. intersects any sufficiently long interval). 2 If ( X, T ) is a distal dynamical system, x ∈ U ⊂ X with U open, then the set { n ∈ N : T n x ∈ U } is IP ∗ . Theorem (Hindman) If A is an IP set, A = A 1 ∪ A 2 ∪ · · · ∪ A r then ∃ j : A j is IP . If B 1 , B 2 . . . , B r are IP ∗ sets then B = B 1 ∩ B 2 ∩ · · · ∩ B r is IP ∗ . 11 / 22

IP sets and GP sets Corollary (Bergelson–Leibman) Let g : Z → R d be a bounded generalised polynomial. Then for almost all n ∈ N , for any δ > 0 , the set { m ∈ N : | g ( n + m ) − g ( n ) | < δ } is IP ∗ . In particular, if A ⊂ N is a GP set then A − n is IP ∗ for almost all n ∈ A . Proof. Represent g ( n ) = F ( T n x 0 ) , as in Bergelson–Leibman Theorem. Because F is piecewise polynomial, F is continuous almost everywhere. In particular, for almost every n , F is continuous at T n x 0 . Restrict to such n . Any nilsystem is distal. Hence, for any open U ∋ T n x 0 , the set m ∈ N : T m + n x 0 ∈ U is IP ∗ . Pick U so that | F ( x ) − F ( y ) | < δ for � � all x, y ∈ U . 12 / 22