Configurations in Lattices & Multiple Mixing Alex Gorodnik - PowerPoint PPT Presentation

Configurations in Lattices & Multiple Mixing Alex Gorodnik (University of Bristol) joint work with Michael Bj¨ orklund and Manfred Einsiedler

Configurations in R d

Configurations in R d Ω = a “large” subset of R d . Question Does Ω contain an isometric copy of a given configuration ∆ = { v 1 , . . . , v k +1 } ⊂ R d ?

Configurations in R d Ω = a “large” subset of R d . Question Does Ω contain an isometric copy of a given configuration ∆ = { v 1 , . . . , v k +1 } ⊂ R d ? Assume that Ω has positive upper density (i.e., lim | Ω ∩ B R | > 0). | B R |

Configurations in R d Ω = a “large” subset of R d . Question Does Ω contain an isometric copy of a given configuration ∆ = { v 1 , . . . , v k +1 } ⊂ R d ? Assume that Ω has positive upper density (i.e., lim | Ω ∩ B R | > 0). | B R | Furstenberg-Katznelson-Weiss, Bourgain, Quas: If k < d and ∆ is a simplex, then Ω contains an isometric copy of the dilation t ∆ for sufficiently large t .

Configurations in R d Ω = a “large” subset of R d . Question Does Ω contain an isometric copy of a given configuration ∆ = { v 1 , . . . , v k +1 } ⊂ R d ? Assume that Ω has positive upper density (i.e., lim | Ω ∩ B R | > 0). | B R | Furstenberg-Katznelson-Weiss, Bourgain, Quas: If k < d and ∆ is a simplex, then Ω contains an isometric copy of the dilation t ∆ for sufficiently large t . Bourgain, Graham: some counterexaples when k ≥ d ,

Configurations in R d Ω = a “large” subset of R d . Question Does Ω contain an isometric copy of a given configuration ∆ = { v 1 , . . . , v k +1 } ⊂ R d ? Assume that Ω has positive upper density (i.e., lim | Ω ∩ B R | > 0). | B R | Furstenberg-Katznelson-Weiss, Bourgain, Quas: If k < d and ∆ is a simplex, then Ω contains an isometric copy of the dilation t ∆ for sufficiently large t . Bourgain, Graham: some counterexaples when k ≥ d , but the case of when ∆ is a triangle in R 2 is still open!

Configurations in R d Ω = a “large” subset of R d . Question Does Ω contain an isometric copy of a given configuration ∆ = { v 1 , . . . , v k +1 } ⊂ R d ? Assume that Ω has positive upper density (i.e., lim | Ω ∩ B R | > 0). | B R | Furstenberg-Katznelson-Weiss, Bourgain, Quas: If k < d and ∆ is a simplex, then Ω contains an isometric copy of the dilation t ∆ for sufficiently large t . Bourgain, Graham: some counterexaples when k ≥ d , but the case of when ∆ is a triangle in R 2 is still open! Furstenberg-Katznelson-Weiss, Ziegler: In general, every ε -neighbourhood of Ω contains an isometric copy of the dilation t ∆ for sufficiently large t .

Configurations in other groups G = a group (with a right-invariant metric), Ω = a “large” subset of G .

Configurations in other groups G = a group (with a right-invariant metric), Ω = a “large” subset of G . Question Does Ω contain an isometric copy of a given configuration ∆ = { g 1 , . . . , g k } ⊂ G?

Configurations in other groups G = a group (with a right-invariant metric), Ω = a “large” subset of G . Question Does Ω contain an isometric copy of a given configuration ∆ = { g 1 , . . . , g k } ⊂ G? Bergelson-McCutcheon-Zhang: Every Ω ⊂ G × G of positive upper density (here G is a countable amenable group) contains many configurations of the form { (1 , 1) , ( g , 1) , ( g , g ) } · h with g ∈ G , h ∈ G × G .

Configurations in other groups G = a group (with a right-invariant metric), Ω = a “large” subset of G . Question Does Ω contain an isometric copy of a given configuration ∆ = { g 1 , . . . , g k } ⊂ G? Bergelson-McCutcheon-Zhang: Every Ω ⊂ G × G of positive upper density (here G is a countable amenable group) contains many configurations of the form { (1 , 1) , ( g , 1) , ( g , g ) } · h with g ∈ G , h ∈ G × G . Furstenberg-Glasner : Given Ω ⊂ SL 2 ( R ) of positive measure (w.r.t. a suitable mean on SL 2 ( R )), every ε -neighbourhood of Ω contains many configurations of the form { g , g 2 , . . . , g k } · h with g , h ∈ SL 2 ( R ).

Configurations in lattice subgroups G = a connected Lie group, Γ = a discrete subgroup of G with finite covolume. Question Does an ε -neighbourhood of Γ contain an isometric copy of a given configuration ∆ = { g 1 , . . . , g k } ⊂ G?

Configurations in lattice subgroups G = a connected Lie group, Γ = a discrete subgroup of G with finite covolume. Question Does an ε -neighbourhood of Γ contain an isometric copy of a given configuration ∆ = { g 1 , . . . , g k } ⊂ G? In particular, which R > 0 can be approximated by d ( γ, e ), γ ∈ Γ?

Example: SL 2 ( Z ) Consider the orbit Γ · i of Γ = SL 2 ( Z ) in the hyperbolic plane H 2 .

Example: SL 2 ( Z ) Consider the orbit Γ · i of Γ = SL 2 ( Z ) in the hyperbolic plane H 2 . � a � b For γ = ∈ Γ, c d d ( γ i , i ) = cosh − 1 ( a 2 + b 2 + c 2 + d 2 ) / 2 .

Example: SL 2 ( Z ) Consider the orbit Γ · i of Γ = SL 2 ( Z ) in the hyperbolic plane H 2 . � a � b For γ = ∈ Γ, c d d ( γ i , i ) = cosh − 1 ( a 2 + b 2 + c 2 + d 2 ) / 2 . One can show that if R ≥ const · log( ε − 1 ) , then there exists γ ∈ Γ such that | R − d ( γ i , i ) | < ε.

Example: SL 2 ( Z ) Consider the orbit Γ · i of Γ = SL 2 ( Z ) in the hyperbolic plane H 2 . � a � b For γ = ∈ Γ, c d d ( γ i , i ) = cosh − 1 ( a 2 + b 2 + c 2 + d 2 ) / 2 . One can show that if R ≥ const · log( ε − 1 ) , then there exists γ ∈ Γ such that | R − d ( γ i , i ) | < ε. However, this fails for R = o (log( ε − 1 ))!

Configurations in lattice subgroups G = a simple connected noncompact Lie group (e.g, G = SL n ( R )), Γ = a discrete subgroup with finite covolume.

Configurations in lattice subgroups G = a simple connected noncompact Lie group (e.g, G = SL n ( R )), Γ = a discrete subgroup with finite covolume. We fix a right-invariant Riemannian metric d ( · , · ) on G .

Configurations in lattice subgroups G = a simple connected noncompact Lie group (e.g, G = SL n ( R )), Γ = a discrete subgroup with finite covolume. We fix a right-invariant Riemannian metric d ( · , · ) on G . Theorem (Bj¨ orklund, Einsiedler, G.) Let ∆ = { g 1 , . . . , g k } be a configuration in G such that d ( g i , g j ) ≥ const · log( ε − 1 ) for i � = j . Then ε -neighbourhood of Γ contains the configuration ∆ · h for some h ∈ G.

Configurations in lattice subgroups G = a simple connected noncompact Lie group (e.g, G = SL n ( R )), Γ = a discrete subgroup with finite covolume. We fix a right-invariant Riemannian metric d ( · , · ) on G . Theorem (Bj¨ orklund, Einsiedler, G.) Let ∆ = { g 1 , . . . , g k } be a configuration in G such that d ( g i , g j ) ≥ const · log( ε − 1 ) for i � = j . Then ε -neighbourhood of Γ contains the configuration ∆ · h for some h ∈ G. Main ingredient of the proof: analysis of higher-order correlations.

Exponential multiple mixing Notation: X = G / Γ with the normalised volume m , � 1 / 2 �� X | D α φ | 2 dm � � φ � ℓ := – the Sobolev norm. | α |≤ ℓ

Exponential multiple mixing Notation: X = G / Γ with the normalised volume m , � 1 / 2 �� X | D α φ | 2 dm � � φ � ℓ := – the Sobolev norm. | α |≤ ℓ Theorem (Bj¨ orklund, Einsiedler, G.) There exists δ > 0 such that for any functions φ 1 , . . . , φ k : X → R in a suitable Sobolev space and any g 1 , . . . , g k ∈ G, �� � �� � � φ 1 ( g 1 x ) · · · φ k ( g k x ) dx = φ 1 dm · · · φ k dm X X X � � e − δ N ( g 1 ,..., g k ) � φ 1 � ℓ · · · � φ k � ℓ + O where N ( g 1 , . . . , g k ) = min i � = j d ( g i , g j ) .

Exponential multiple mixing Notation: X = G / Γ with the normalised volume m , � 1 / 2 �� X | D α φ | 2 dm � � φ � ℓ := – the Sobolev norm. | α |≤ ℓ Theorem (Bj¨ orklund, Einsiedler, G.) There exists δ > 0 such that for any functions φ 1 , . . . , φ k : X → R in a suitable Sobolev space and any g 1 , . . . , g k ∈ G, �� � �� � � φ 1 ( g 1 x ) · · · φ k ( g k x ) dx = φ 1 dm · · · φ k dm X X X � � e − δ N ( g 1 ,..., g k ) � φ 1 � ℓ · · · � φ k � ℓ + O where N ( g 1 , . . . , g k ) = min i � = j d ( g i , g j ) . Borel-Wallach, Cowling, Howe-Moore: exponential 2-mixing,

Exponential multiple mixing Notation: X = G / Γ with the normalised volume m , � 1 / 2 �� X | D α φ | 2 dm � � φ � ℓ := – the Sobolev norm. | α |≤ ℓ Theorem (Bj¨ orklund, Einsiedler, G.) There exists δ > 0 such that for any functions φ 1 , . . . , φ k : X → R in a suitable Sobolev space and any g 1 , . . . , g k ∈ G, �� � �� � � φ 1 ( g 1 x ) · · · φ k ( g k x ) dx = φ 1 dm · · · φ k dm X X X � � e − δ N ( g 1 ,..., g k ) � φ 1 � ℓ · · · � φ k � ℓ + O where N ( g 1 , . . . , g k ) = min i � = j d ( g i , g j ) . Borel-Wallach, Cowling, Howe-Moore: exponential 2-mixing, Mozes: multiple mixing without quantitative estimate,