Equidistribution for groups of toral automorphisms

1/17 Equidistribution for groups of toral automorphisms J. Bourgain A. Furman E. Lindenstrauss S. Mozes 1 Institute for Advanced Study 2 University of Illinois at Chicago 3 Princeton and Hebrew University in Jerusalem 4 Hebrew University in Jerusalem UIC, May 2010

2/17 Basic dynamical questions General goal T : X X homeomorphism of a compact space X Understand the distribution of x, Tx,..., T N x as N.

2/17 Basic dynamical questions General goal T : X X homeomorphism of a compact space X Understand the distribution of x, Tx,..., T N x as N. Levels of understanding Equidistribution: x X, µ x P T (X ) N 1 1 f (T n x) N n=0 X f (y) dµ x (y) (f C(X ))

2/17 Basic dynamical questions General goal T : X X homeomorphism of a compact space X Understand the distribution of x, Tx,..., T N x as N. Levels of understanding Equidistribution: x X, µ x P T (X ) N 1 1 f (T n x) N n=0 X f (y) dµ x (y) (f C(X )) Invariant measures: P T (X ) = {µ P(X ) : T µ = µ}

2/17 Basic dynamical questions General goal T : X X homeomorphism of a compact space X Understand the distribution of x, Tx,..., T N x as N. Levels of understanding Equidistribution: x X, µ x P T (X ) N 1 1 f (T n x) N n=0 X f (y) dµ x (y) (f C(X )) Invariant measures: P T (X ) = {µ P(X ) : T µ = µ} Closed Invariant sets

Toral automorphisms A SL d (Z) acts on T d = R d /Z d by A : x + Z d Ax + Z d 3/17

3/17 Toral automorphisms A SL d (Z) acts on T d = R d /Z d by A : x + Z d Ax + Z d Standard Example A = ( 2 1 1 1 )

3/17 Toral automorphisms A SL d (Z) acts on T d = R d /Z d by A : x + Z d Ax + Z d Standard Example A = ( 2 1 1 1 ) Observation {( ) } Periodic points = p1 q,..., p d q + Z d : gcd(p 1,..., p d, q) = 1

3/17 Toral automorphisms A SL d (Z) acts on T d = R d /Z d by A : x + Z d Ax + Z d Standard Example A = ( 2 1 1 1 ) Observation {( ) } Periodic points = p1 q,..., p d q + Z d : gcd(p 1,..., p d, q) = 1 Single hyperbolic automorphism 1 Closed Invariant sets: of every Hausdorff dim [0, d]

3/17 Toral automorphisms A SL d (Z) acts on T d = R d /Z d by A : x + Z d Ax + Z d Standard Example A = ( 2 1 1 1 ) Observation {( ) } Periodic points = p1 q,..., p d q + Z d : gcd(p 1,..., p d, q) = 1 Single hyperbolic automorphism 1 Closed Invariant sets: of every Hausdorff dim [0, d] 2 Invariant measures: uncountably many distinct ergodic

3/17 Toral automorphisms A SL d (Z) acts on T d = R d /Z d by A : x + Z d Ax + Z d Standard Example A = ( 2 1 1 1 ) Observation {( ) } Periodic points = p1 q,..., p d q + Z d : gcd(p 1,..., p d, q) = 1 Single hyperbolic automorphism 1 Closed Invariant sets: of every Hausdorff dim [0, d] 2 Invariant measures: uncountably many distinct ergodic 3 Equidistribution: no chance!

4/17 Abelian groups of toral automorphisms Setup Non degenerate Z k < SL d (Z) with 2 k d 1

4/17 Abelian groups of toral automorphisms Setup Non degenerate Z k < SL d (Z) with 2 k d 1 Rigidity phenomena

4/17 Abelian groups of toral automorphisms Setup Non degenerate Z k < SL d (Z) with 2 k d 1 Rigidity phenomena 1 Closed Invariant sets: Finite (rational pts), T d H. Furstenberg (77), D. Berend (84)

4/17 Abelian groups of toral automorphisms Setup Non degenerate Z k < SL d (Z) with 2 k d 1 Rigidity phenomena 1 Closed Invariant sets: Finite (rational pts), T d H. Furstenberg (77), D. Berend (84) 2 Invariant measures: Conjecture: Atomic (rational pts) + Lebesgue

4/17 Abelian groups of toral automorphisms Setup Non degenerate Z k < SL d (Z) with 2 k d 1 Rigidity phenomena 1 Closed Invariant sets: Finite (rational pts), T d H. Furstenberg (77), D. Berend (84) 2 Invariant measures: Conjecture: Atomic (rational pts) + Lebesgue Positive entropy (equivalently dimh (µ) > 0) understood by: D. Rudolph, A. Katok, R. Spatzier, B. Host, B. Kalinin, E. Lindenstrauss, M. Einsiedler,...

4/17 Abelian groups of toral automorphisms Setup Non degenerate Z k < SL d (Z) with 2 k d 1 Rigidity phenomena 1 Closed Invariant sets: Finite (rational pts), T d H. Furstenberg (77), D. Berend (84) 2 Invariant measures: Conjecture: Atomic (rational pts) + Lebesgue Positive entropy (equivalently dimh (µ) > 0) understood by: D. Rudolph, A. Katok, R. Spatzier, B. Host, B. Kalinin, E. Lindenstrauss, M. Einsiedler,... 3 No equidistribution

5/17 Large groups of toral automorphisms Setup Γ < SL d (Z) which is Zariski dense in SL d (R)

5/17 Large groups of toral automorphisms Setup Γ < SL d (Z) which is Zariski dense in SL d (R) What is equidistribution for Γ.x?

5/17 Large groups of toral automorphisms Setup Γ < SL d (Z) which is Zariski dense in SL d (R) What is equidistribution for Γ.x? Fix a prob meas ν on Γ with Γ = supp(ν). Consider µ n,x = ν n δ x = ν(g n ) ν(g 1 ) δ gn g 1x.

5/17 Large groups of toral automorphisms Setup Γ < SL d (Z) which is Zariski dense in SL d (R) What is equidistribution for Γ.x? Fix a prob meas ν on Γ with Γ = supp(ν). Consider µ n,x = ν n δ x = ν(g n ) ν(g 1 ) δ gn g 1x. Remark 1 N 1 Weak-* limits of N n=0 µ n,x are ν-stationary measures { P ν (X ) = µ P(X ) : µ = ν µ = } ν(g) g µ

5/17 Large groups of toral automorphisms Setup Γ < SL d (Z) which is Zariski dense in SL d (R) What is equidistribution for Γ.x? Fix a prob meas ν on Γ with Γ = supp(ν). Consider µ n,x = ν n δ x = ν(g n ) ν(g 1 ) δ gn g 1x. Remark 1 N 1 Weak-* limits of N n=0 µ n,x are ν-stationary measures { P ν (X ) = µ P(X ) : µ = ν µ = } ν(g) g µ P Γ (X ) P ν (X ) convex compact subsets of P(X )

5/17 Large groups of toral automorphisms Setup Γ < SL d (Z) which is Zariski dense in SL d (R) What is equidistribution for Γ.x? Fix a prob meas ν on Γ with Γ = supp(ν). Consider µ n,x = ν n δ x = ν(g n ) ν(g 1 ) δ gn g 1x. Remark 1 N 1 Weak-* limits of N n=0 µ n,x are ν-stationary measures { P ν (X ) = µ P(X ) : µ = ν µ = } ν(g) g µ P Γ (X ) P ν (X ) convex compact subsets of P(X ) P Γ (X ) = is possible for non-amenable Γ.

Large groups of toral automorphisms Setup Γ < SL d (Z) which is Zariski dense in SL d (R) What is equidistribution for Γ.x? Fix a prob meas ν on Γ with Γ = supp(ν). Consider µ n,x = ν n δ x = ν(g n ) ν(g 1 ) δ gn g 1x. Remark 1 N 1 Weak-* limits of N n=0 µ n,x are ν-stationary measures { P ν (X ) = µ P(X ) : µ = ν µ = } ν(g) g µ P Γ (X ) P ν (X ) convex compact subsets of P(X ) P Γ (X ) = is possible for non-amenable Γ. P ν (X ), any closed invariant set supports ν-stationary measures 5/17

6/17 Overview of the results Setup Γ < SL d (Z) which is Z-dense, or more generally

6/17 Overview of the results Setup Γ < SL d (Z) which is Z-dense, or more generally Γ strongly irreducible and Γ proximal element

6/17 Overview of the results Setup Γ < SL d (Z) which is Z-dense, or more generally Γ strongly irreducible and Γ proximal element Rigidity phenomena 1 Closed Γ-invariant sets = Finite (rational pts), T d R. Muchnik (05), Y. Guivarc h-a. Starkov (04)

6/17 Overview of the results Setup Γ < SL d (Z) which is Z-dense, or more generally Γ strongly irreducible and Γ proximal element Rigidity phenomena 1 Closed Γ-invariant sets = Finite (rational pts), T d R. Muchnik (05), Y. Guivarc h-a. Starkov (04) 2 Γ-invariant measures = Atomic (rational pts) + Lebesgue BFLM (07, 10), Y. Benoist-J.F. Quint (10)

6/17 Overview of the results Setup Γ < SL d (Z) which is Z-dense, or more generally Γ strongly irreducible and Γ proximal element Rigidity phenomena 1 Closed Γ-invariant sets = Finite (rational pts), T d R. Muchnik (05), Y. Guivarc h-a. Starkov (04) 2 Γ-invariant measures = Atomic (rational pts) + Lebesgue BFLM (07, 10), Y. Benoist-J.F. Quint (10) 3 ν-stationary measures = Γ-invariant = Atomic + Lebesgue BLFM (07, 10), Y. Benoist-J.F. Quint (10)

6/17 Overview of the results Setup Γ < SL d (Z) which is Z-dense, or more generally Γ strongly irreducible and Γ proximal element Rigidity phenomena 1 Closed Γ-invariant sets = Finite (rational pts), T d R. Muchnik (05), Y. Guivarc h-a. Starkov (04) 2 Γ-invariant measures = Atomic (rational pts) + Lebesgue BFLM (07, 10), Y. Benoist-J.F. Quint (10) 3 ν-stationary measures = Γ-invariant = Atomic + Lebesgue BLFM (07, 10), Y. Benoist-J.F. Quint (10) 4 Equidistribution (in fact, quantitative!) BLFM (07, 10). [BFLM] Stationary measures and equidistribution for orbits of non-abelian semi-groups on the torus, JAMS to appear.

The main result (BFLM) Assume ν on SL d (Z) with Γ = supp(ν) str irr + prox elmt and 7/17

7/17 The main result (BFLM) Assume ν on SL d (Z) with Γ = supp(ν) str irr + prox elmt and ɛ > 0 g ν(g) g ɛ <.

7/17 The main result (BFLM) Assume ν on SL d (Z) with Γ = supp(ν) str irr + prox elmt and ɛ > 0 g ν(g) g ɛ <. Theorem (BFLM) 1 If x T d is irrational then µ n,x = ν n δ x Leb

7/17 The main result (BFLM) Assume ν on SL d (Z) with Γ = supp(ν) str irr + prox elmt and ɛ > 0 g ν(g) g ɛ <. Theorem (BFLM) 1 If x T d is irrational then µ n,x = ν n δ x Leb 2 If x T d is M-Diophantine ( x p q > 1 q M ) then

7/17 The main result (BFLM) Assume ν on SL d (Z) with Γ = supp(ν) str irr + prox elmt and ɛ > 0 g ν(g) g ɛ <. Theorem (BFLM) 1 If x T d is irrational then µ n,x = ν n δ x Leb 2 If x T d is M-Diophantine ( x p q > 1 q M ) then µ n,x (a) < a e cn/m

7/17 The main result (BFLM) Assume ν on SL d (Z) with Γ = supp(ν) str irr + prox elmt and ɛ > 0 g ν(g) g ɛ <. Theorem (BFLM) 1 If x T d is irrational then µ n,x = ν n δ x Leb 2 If x T d is M-Diophantine ( x p q > 1 q M ) then µ n,x (a) < a e cn/m 3 If µ n,x (a) = t > 0 for some a Z d {0} with n > C log(2 a /t)

7/17 The main result (BFLM) Assume ν on SL d (Z) with Γ = supp(ν) str irr + prox elmt and ɛ > 0 g ν(g) g ɛ <. Theorem (BFLM) 1 If x T d is irrational then µ n,x = ν n δ x Leb 2 If x T d is M-Diophantine ( x p q > 1 q M ) then µ n,x (a) < a e cn/m 3 If µ n,x (a) = t > 0 for some a Z d {0} with n > C log(2 a /t) then x p ( ) C 2 a q < e λn with q < t where c > 0, λ > 0, C depend only on ν, µ(a) = e 2πi a,x dµ(x) for a Z d. T d

8/17 Baby case Theorem (M. Burger) Let µ P(T d ) be invariant under a finite index subgroup Γ < SL d (Z). Then µ is a convex combination of Leb and atomic on finite orbits.

8/17 Baby case Theorem (M. Burger) Let µ P(T d ) be invariant under a finite index subgroup Γ < SL d (Z). Then µ is a convex combination of Leb and atomic on finite orbits. Proof 1 ĝ µ(a) = T d e 2πi a,gx dµ(x) = T d e 2πi g tr a,x dµ(x) = µ(g tr a)

8/17 Baby case Theorem (M. Burger) Let µ P(T d ) be invariant under a finite index subgroup Γ < SL d (Z). Then µ is a convex combination of Leb and atomic on finite orbits. Proof 1 ĝ µ(a) = e 2πi a,gx dµ(x) = e 2πi g tr a,x dµ(x) = µ(g tr a) T d T d 2 Wiener s Lemma: x T d µ({x}) 2 = lim 1 B n a B n µ(a) 2

8/17 Baby case Theorem (M. Burger) Let µ P(T d ) be invariant under a finite index subgroup Γ < SL d (Z). Then µ is a convex combination of Leb and atomic on finite orbits. Proof 1 ĝ µ(a) = e 2πi a,gx dµ(x) = e 2πi g tr a,x dµ(x) = µ(g tr a) T d T d 2 Wiener s Lemma: x T d µ({x}) 2 = lim 1 B n a B n µ(a) 2 Assume µ is Γ-invariant and µ Leb. µ(a) = t > 0 for some a Z d \ {0}

8/17 Baby case Theorem (M. Burger) Let µ P(T d ) be invariant under a finite index subgroup Γ < SL d (Z). Then µ is a convex combination of Leb and atomic on finite orbits. Proof 1 ĝ µ(a) = e 2πi a,gx dµ(x) = e 2πi g tr a,x dµ(x) = µ(g tr a) T d T d 2 Wiener s Lemma: x T d µ({x}) 2 = lim 1 B n a B n µ(a) 2 Assume µ is Γ-invariant and µ Leb. µ(a) = t > 0 for some a Z d \ {0} µ(g tr a) 2 = µ(a) 2 = t 2 (g Γ)

8/17 Baby case Theorem (M. Burger) Let µ P(T d ) be invariant under a finite index subgroup Γ < SL d (Z). Then µ is a convex combination of Leb and atomic on finite orbits. Proof 1 ĝ µ(a) = e 2πi a,gx dµ(x) = e 2πi g tr a,x dµ(x) = µ(g tr a) T d T d 2 Wiener s Lemma: x T d µ({x}) 2 = lim 1 B n a B n µ(a) 2 Assume µ is Γ-invariant and µ Leb. µ(a) = t > 0 for some a Z d \ {0} µ(g tr a) 2 = µ(a) 2 = t 2 (g Γ) Density( µ 2 ) t 2 Density(Γ tr.a) > 0

8/17 Baby case Theorem (M. Burger) Let µ P(T d ) be invariant under a finite index subgroup Γ < SL d (Z). Then µ is a convex combination of Leb and atomic on finite orbits. Proof 1 ĝ µ(a) = e 2πi a,gx dµ(x) = e 2πi g tr a,x dµ(x) = µ(g tr a) T d T d 2 Wiener s Lemma: x T d µ({x}) 2 = lim 1 B n a B n µ(a) 2 Assume µ is Γ-invariant and µ Leb. µ(a) = t > 0 for some a Z d \ {0} µ(g tr a) 2 = µ(a) 2 = t 2 (g Γ) Density( µ 2 ) t 2 Density(Γ tr.a) > 0 µ has atoms (by Wiener)

8/17 Baby case Theorem (M. Burger) Let µ P(T d ) be invariant under a finite index subgroup Γ < SL d (Z). Then µ is a convex combination of Leb and atomic on finite orbits. Proof 1 ĝ µ(a) = e 2πi a,gx dµ(x) = e 2πi g tr a,x dµ(x) = µ(g tr a) T d T d 2 Wiener s Lemma: x T d µ({x}) 2 = lim 1 B n a B n µ(a) 2 Assume µ is Γ-invariant and µ Leb. µ(a) = t > 0 for some a Z d \ {0} µ(g tr a) 2 = µ(a) 2 = t 2 (g Γ) Density( µ 2 ) t 2 Density(Γ tr.a) > 0 µ has atoms (by Wiener) Atoms of a Γ-inv prob measure belong to finite orbits.

9/17 First glance at the problem Make the proof for the following effective If µ = ν µ has µ(a) = t > 0 for some a Z d \ {0}. Then µ has atoms.

9/17 First glance at the problem Make the proof for the following effective If µ = ν µ has µ(a) = t > 0 for some a Z d \ {0}. Then µ has atoms. Difficulties 1 ˆµ is not constant on Γ-orbits

9/17 First glance at the problem Make the proof for the following effective If µ = ν µ has µ(a) = t > 0 for some a Z d \ {0}. Then µ has atoms. Difficulties 1 ˆµ is not constant on Γ-orbits 2 Γ-orbits on Z d have zero density

9/17 First glance at the problem Make the proof for the following effective If µ = ν µ has µ(a) = t > 0 for some a Z d \ {0}. Then µ has atoms. Difficulties 1 ˆµ is not constant on Γ-orbits 2 Γ-orbits on Z d have zero density Overcoming the difficulties 1 µ = ν µ = = ν n µ µ(a) = ν n (g) µ(g tr a)

9/17 First glance at the problem Make the proof for the following effective If µ = ν µ has µ(a) = t > 0 for some a Z d \ {0}. Then µ has atoms. Difficulties 1 ˆµ is not constant on Γ-orbits 2 Γ-orbits on Z d have zero density Overcoming the difficulties 1 µ = ν µ = = ν n µ µ(a) = ν n (g) µ(g tr a) So µ(a) > t ν n { g : µ(g tr a) > 1 2 t} > 1 2 t

9/17 First glance at the problem Make the proof for the following effective If µ = ν µ has µ(a) = t > 0 for some a Z d \ {0}. Then µ has atoms. Difficulties 1 ˆµ is not constant on Γ-orbits 2 Γ-orbits on Z d have zero density Overcoming the difficulties 1 µ = ν µ = = ν n µ µ(a) = ν n (g) µ(g tr a) So µ(a) > t ν n { g : µ(g tr a) > 1 2 t} > 1 2 t 2 This is 99% of the work!

General strategy 10/17

10/17 General strategy 1 Density> 0 at scales N, M = N 1 κ for A s = {b Z d : µ(b) > s} N M (A c1(t) [ N, N] d ) > c 2 (t) ( N M where N M (S) - minimal number of M-cubes needed to cover S ) d

General strategy 1 Density> 0 at scales N, M = N 1 κ for A s = {b Z d : µ(b) > s} N M (A c1(t) [ N, N] d ) > c 2 (t) ( N M where N M (S) - minimal number of M-cubes needed to cover S N ) d S = 12 N M (S) = 5

10/17 General strategy 1 Density> 0 at scales N, M = N 1 κ for A s = {b Z d : µ(b) > s} N M (A c1(t) [ N, N] d ) > c 2 (t) ( N M where N M (S) - minimal number of M-cubes needed to cover S ) d M N S = 12 N M (S) = 5

11/17 General strategy 1 Density> 0 at scales N, M = N 1 κ for A s = {b Z d : µ(b) > s} 2 µ is granulated N M (A c1(t) [ N, N] d ) > c 2 (t) ( N M ) d

11/17 General strategy 1 Density> 0 at scales N, M = N 1 κ for A s = {b Z d : µ(b) > s} N M (A c1(t) [ N, N] d ) > c 2 (t) ( N M 2 µ is granulated There is 1/M-separated set {x 1,..., x M d ( } T d M ) d µ > c 3 (t), µ(b xi,r ) > r d(1 κ) i=1 B x i, 1 N ) d

11/17 General strategy 1 Density> 0 at scales N, M = N 1 κ for A s = {b Z d : µ(b) > s} N M (A c1(t) [ N, N] d ) > c 2 (t) ( N M 2 µ is granulated There is 1/M-separated set {x 1,..., x M d ( } T d M ) d µ > c 3 (t), µ(b xi,r ) > r d(1 κ) i=1 B x i, 1 N 3 From granulation to atoms at rational points: Positive µ-mass at very dense balls µ(by,ρ) > ρ ɛ ) d

11/17 General strategy 1 Density> 0 at scales N, M = N 1 κ for A s = {b Z d : µ(b) > s} N M (A c1(t) [ N, N] d ) > c 2 (t) ( N M 2 µ is granulated There is 1/M-separated set {x 1,..., x M d ( } T d M ) d µ > c 3 (t), µ(b xi,r ) > r d(1 κ) i=1 B x i, 1 N 3 From granulation to atoms at rational points: Positive µ-mass at very dense balls µ(by,ρ) > ρ ɛ Dense balls are attracted to rational points ) d

11/17 General strategy 1 Density> 0 at scales N, M = N 1 κ for A s = {b Z d : µ(b) > s} N M (A c1(t) [ N, N] d ) > c 2 (t) ( N M 2 µ is granulated There is 1/M-separated set {x 1,..., x M d ( } T d M ) d µ > c 3 (t), µ(b xi,r ) > r d(1 κ) i=1 B x i, 1 N 3 From granulation to atoms at rational points: Positive µ-mass at very dense balls µ(by,ρ) > ρ ɛ Dense balls are attracted to rational points Gravitational collapse ) d

Products of random matrices G = KA + K: g = k diag[e t1,..., e t d ] k k, k SO(d), t 1 t d A

Products of random matrices G = KA + K: g = k diag[e t1,..., e t d ] k k, k SO(d), t 1 t d A k(a)

Products of random matrices G = KA + K: g = k diag[e t1,..., e t d ] k k, k SO(d), t 1 t d e t e t A k(a)

Products of random matrices G = KA + K: g = k diag[e t1,..., e t d ] k k, k SO(d), t 1 t d A k(a) ( e t 0 0 e t e t e t ) k(a)

12/17 Products of random matrices G = KA + K: g = k diag[e t1,..., e t d ] k k, k SO(d), t 1 t d θ A k(a) ( e t 0 0 e t e t e t ) k(a) g(a)

12/17 Products of random matrices G = KA + K: g = k diag[e t1,..., e t d ] k k, k SO(d), t 1 t d θ A k(a) ( e t 0 0 e t e t e t ) k(a) g(a) Furstenberg, Guivarc h, Raugi, LaPage, Goldsheid-Margulis,...

12/17 Products of random matrices G = KA + K: g = k diag[e t1,..., e t d ] k k, k SO(d), t 1 t d θ A k(a) ( e t 0 0 e t e t e t ) k(a) g(a) Furstenberg, Guivarc h, Raugi, LaPage, Goldsheid-Margulis,... { ( ) } e ν n g = k (λ±ɛ)n 0 0 e ( λ±ɛ)n k > 1 e cn

12/17 Products of random matrices G = KA + K: g = k diag[e t1,..., e t d ] k k, k SO(d), t 1 t d θ A k(a) ( e t 0 0 e t e t e t ) k(a) g(a) Furstenberg, Guivarc h, Raugi, LaPage, Goldsheid-Margulis,... { ( ) } e ν n g = k (λ±ɛ)n 0 0 e ( λ±ɛ)n k > 1 e cn ν n {g θ g B ξ,r } < r γ for e cn < r

Large scale dimension 13/17

13/17 Given µ(a 0 ) = t 0 > 0 Large scale dimension

13/17 Given µ(a 0 ) = t 0 > 0 Large scale dimension A s = {b Z d : µ(b) > s}

13/17 Given µ(a 0 ) = t 0 > 0 Large scale dimension A s = {b Z d : µ(b) > s} Want to show N M (A c(t) B 0,N ) > c(t) ( N M ) d

13/17 Given µ(a 0 ) = t 0 > 0 Large scale dimension A s = {b Z d : µ(b) > s} Want to show N M (A c(t) B 0,N ) > c(t) ( N M ) d Random walks + stationarity ν n {g : µ(g tr a 0 ) > 1 2 t 0} > 1 2 t 0

13/17 Given µ(a 0 ) = t 0 > 0 Large scale dimension A s = {b Z d : µ(b) > s} Want to show N M (A c(t) B 0,N ) > c(t) ( N M ) d Random walks + stationarity ν n {g : µ(g tr a 0 ) > 1 2 t 0} > 1 2 t 0 gives α 0 > 0 N M (A c0(t 0) B 0,N ) > c 0 (t) ( N M ) α0

13/17 Given µ(a 0 ) = t 0 > 0 Large scale dimension A s = {b Z d : µ(b) > s} Want to show N M (A c(t) B 0,N ) > c(t) ( N M ) d Random walks + stationarity ν n {g : µ(g tr a 0 ) > 1 2 t 0} > 1 2 t 0 gives α 0 > 0 N M (A c0(t 0) B 0,N ) > c 0 (t) ( N M Need to improve the dimension α = α 0 to α = d in steps α i α i+1 ( ) αi Ni N Mi (A ti B 0,Ni ) > c i (t) M i ) α0

14/17 Additive structure of Fourier coefficients Lemma 1 A 2 a,b A µ(a b) 1 A a A µ(a) 2

14/17 Additive structure of Fourier coefficients Lemma 1 A 2 a,b A µ(a b) 1 A a A µ(a) 2 Proof. 1 1 A 2 a,b A µ(a b) = = ( ) 1 T a A d e2πi a,x A A 2 a,b A ( 1 A T d e 2πi a b,x dµ(x) b A e2πi b,x ) dµ(x)

14/17 Additive structure of Fourier coefficients Lemma 1 A 2 a,b A µ(a b) 1 A a A µ(a) 2 Proof. 1 1 A 2 a,b A µ(a b) = = ( ) 1 T a A d e2πi a,x A A 2 a,b A ( 1 A T d e 2πi a b,x dµ(x) b A e2πi b,x ) = 1 2 T d A a A e2πi a,x dµ(x) 1 T d A = 1 A a A µ(a) 2 dµ(x) a A e2πi a,x dµ(x) 2

Bourgain s Projection Theorem (informal) 15/17

15/17 Bourgain s Projection Theorem (informal) A R d, dim(a) = α, θ Θ Θ P d 1, dim(θ) γ dim(π θ (A)) > α+δ d

15/17 Bourgain s Projection Theorem (informal) A R d, dim(a) = α, θ Θ Θ P d 1, dim(θ) γ dim(π θ (A)) > α+δ d Theorem (Bourgain) β, γ > 0, δ > 0 so that α [β, d β] η P(P d 1 ) with η(b ξ,r ) < r γ A B 0,1 with N r (A) r α A, η not too degenerate Then for η-most θ P d 1 s.t. N r (π θ (A)) > r α+δ d θ A π θ (A)

15/17 Bourgain s Projection Theorem (informal) A R d, dim(a) = α, θ Θ Θ P d 1, dim(θ) γ dim(π θ (A)) > α+δ d Theorem (Bourgain) β, γ > 0, δ > 0 so that α [β, d β] η P(P d 1 ) with η(b ξ,r ) < r γ A B 0,1 with N r (A) r α A, η not too degenerate Then for η-most θ P d 1 s.t. N r (π θ (A)) > r α+δ d Theorem (Marstrand, Falconer) dim A + dim η > d θ, Leb(π θ (A)) > 0 θ A π θ (A)

15/17 Bourgain s Projection Theorem (informal) A R d, dim(a) = α, θ Θ Θ P d 1, dim(θ) γ dim(π θ (A)) > α+δ d Theorem (Bourgain) β, γ > 0, δ > 0 so that α [β, d β] η P(P d 1 ) with η(b ξ,r ) < r γ A B 0,1 with N r (A) r α A, η not too degenerate Then for η-most θ P d 1 s.t. N r (π θ (A)) > r α+δ d Theorem (Marstrand, Falconer) dim A + dim η > d θ, Leb(π θ (A)) > 0 α + γ > d θ, N r (π θ (A)) > cr 1 θ A π θ (A)

A = A ti B 0,Ni dim = α i Amplifying the dimension

Amplifying the dimension A = A ti B 0,Ni dim = α i dim(g tr (A)) > α+δ 2

Amplifying the dimension dim(h tr (A)) > α i +δ 2 A = A ti B 0,Ni dim = α i dim(g tr (A)) > α+δ 2

16/17 Amplifying the dimension dim(h tr (A)) > α i +δ 2 A = A ti B 0,Ni dim = α i g, h : a,b dim(g tr (A)) > α+δ 2 1 A 2 µ(g tr a h tr 1 b) ν(g) µ(g tr a) A g a 2

16/17 Amplifying the dimension dim(h tr (A)) > α i +δ 2 dim(a i+1 ) = α i+1 > α i + δ A = A ti B 0,Ni dim = α i g, h : a,b dim(g tr (A)) > α+δ 2 1 A 2 µ(g tr a h tr 1 b) ν(g) µ(g tr a) A g a 2

Self packing of dense balls

Self packing of dense balls

Self packing of dense balls

Self packing of dense balls 17/17