[PPT] - Rational Approximation on Spheres Dmitry Kleinbock Keith Merrill PowerPoint Presentation

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Classical Results Sn Reduction to dynamics Proofs General Case

Rational Approximation on Spheres

Dmitry Kleinbock Keith Merrill Brandeis University http://arxiv.org/abs/1301.0989 June 4, 2013

June 4, 2013 Heraklion, Crete

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Classical Results Sn Reduction to dynamics Proofs General Case

Motivation and Classical Results

Theorem (Dirichlet’s Theorem on Simultaneous Approximation) For any N > 1 and every x ∈ Rd, there exists p ∈ Zd and q ∈ N such that q ≤ N and

x − p

q

<

1 qN1/d .

June 4, 2013 Heraklion, Crete

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Classical Results Sn Reduction to dynamics Proofs General Case

Motivation and Classical Results

Theorem (Dirichlet’s Theorem on Simultaneous Approximation) For any N > 1 and every x ∈ Rd, there exists p ∈ Zd and q ∈ N such that q ≤ N and

x − p

q

<

1 qN1/d .

June 4, 2013 Heraklion, Crete

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Classical Results Sn Reduction to dynamics Proofs General Case

Motivation and Classical Results

Theorem (Dirichlet’s Theorem on Simultaneous Approximation) For any N > 1 and every x ∈ Rd, there exists p ∈ Zd and q ∈ N such that q ≤ N and

x − p

q

<

1 qN1/d .

June 4, 2013 Heraklion, Crete

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Classical Results Sn Reduction to dynamics Proofs General Case

Motivation and Classical Results

Theorem (Dirichlet’s Theorem on Simultaneous Approximation) For any N > 1 and every x ∈ Rd, there exists p ∈ Zd and q ∈ N such that q ≤ N and

x − p

q

<

1 qN1/d .

June 4, 2013 Heraklion, Crete

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Classical Results Sn Reduction to dynamics Proofs General Case

Motivation and Classical Results

Corollary For every x ∈ Rd ∃∞ p

q ∈ Rd such that

x − p

q

<

1 q1+1/d .

June 4, 2013 Heraklion, Crete

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Classical Results Sn Reduction to dynamics Proofs General Case

Motivation and Classical Results

Corollary For every x ∈ Rd ∃∞ p

q ∈ Rd such that

x − p

q

<

1 q1+1/d .

June 4, 2013 Heraklion, Crete

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Classical Results Sn Reduction to dynamics Proofs General Case

Motivation and Classical Results

Define BA :=

x ∈ Rd : ∃ c > 0 s.t.
x − p

q

>

c q1+1/d ∀ p q ∈ Rd

.

Theorem (Jarník) BA has full Hausdorff dimension. In fact even stronger, it is winning for Schmidt’s game.

June 4, 2013 Heraklion, Crete

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Classical Results Sn Reduction to dynamics Proofs General Case

Motivation and Classical Results

Define BA :=

x ∈ Rd : ∃ c > 0 s.t.
x − p

q

>

c q1+1/d ∀ p q ∈ Rd

.

Theorem (Jarník) BA has full Hausdorff dimension. In fact even stronger, it is winning for Schmidt’s game.

June 4, 2013 Heraklion, Crete

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Classical Results Sn Reduction to dynamics Proofs General Case

Motivation and Classical Results

Define BA :=

x ∈ Rd : ∃ c > 0 s.t.
x − p

q

>

c q1+1/d ∀ p q ∈ Rd

.

Theorem (Jarník) BA has full Hausdorff dimension. In fact even stronger, it is winning for Schmidt’s game.

June 4, 2013 Heraklion, Crete

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Classical Results Sn Reduction to dynamics Proofs General Case

Motivation and Classical Results

For ϕ : N → (0, ∞), we define the set of ϕ-approximable points A(ϕ) :=

x ∈ Rd : ∃∞ p

q s.t.

x − p

q

< ϕ(q)
.

Theorem (Khintchine) Let ϕ : N → (0, ∞) such that k → kϕ(k) is non-increasing. Then m

A(ϕ)
=
if

k kdϕ(k)d < ∞,

∞ if

k kdϕ(k)d = ∞.

Here m denotes Lebesgue measure.

June 4, 2013 Heraklion, Crete

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Classical Results Sn Reduction to dynamics Proofs General Case

Motivation and Classical Results

For ϕ : N → (0, ∞), we define the set of ϕ-approximable points A(ϕ) :=

x ∈ Rd : ∃∞ p

q s.t.

x − p

q

< ϕ(q)
.

Theorem (Khintchine) Let ϕ : N → (0, ∞) such that k → kϕ(k) is non-increasing. Then m

A(ϕ)
=
if

k kdϕ(k)d < ∞,

∞ if

k kdϕ(k)d = ∞.

Here m denotes Lebesgue measure.

June 4, 2013 Heraklion, Crete

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Classical Results Sn Reduction to dynamics Proofs General Case

Motivation and Classical Results

For ϕ : N → (0, ∞), we define the set of ϕ-approximable points A(ϕ) :=

x ∈ Rd : ∃∞ p

q s.t.

x − p

q

< ϕ(q)
.

Theorem (Khintchine) Let ϕ : N → (0, ∞) such that k → kϕ(k) is non-increasing. Then m

A(ϕ)
=
if

k kdϕ(k)d < ∞,

∞ if

k kdϕ(k)d = ∞.

Here m denotes Lebesgue measure.

June 4, 2013 Heraklion, Crete

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Classical Results Sn Reduction to dynamics Proofs General Case

Intrinsic Approximation on Sn

The goal: quantify the density of Sn ∩ Qn+1 in Sn (as opposed to approximating x ∈ Sn by rational points of Rn+1) Past results:

[Schmutz 2008] ∀ N > 1 ∀ x ∈ Sn ∃ p

q ∈ Sn with

q ≤ N and

x − p

q

< 4

√ 2⌈log2(n + 1)⌉ N

1 2⌈log2(n+1)⌉

.

June 4, 2013 Heraklion, Crete

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Classical Results Sn Reduction to dynamics Proofs General Case

Intrinsic Approximation on Sn

The goal: quantify the density of Sn ∩ Qn+1 in Sn (as opposed to approximating x ∈ Sn by rational points of Rn+1) Past results:

[Schmutz 2008] ∀ N > 1 ∀ x ∈ Sn ∃ p

q ∈ Sn with

q ≤ N and

x − p

q

< 4

√ 2⌈log2(n + 1)⌉ N

1 2⌈log2(n+1)⌉

.

June 4, 2013 Heraklion, Crete

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Classical Results Sn Reduction to dynamics Proofs General Case

Intrinsic Approximation on Sn

The goal: quantify the density of Sn ∩ Qn+1 in Sn (as opposed to approximating x ∈ Sn by rational points of Rn+1) Past results:

[Schmutz 2008] ∀ N > 1 ∀ x ∈ Sn ∃ p

q ∈ Sn with

q ≤ N and

x − p

q

< 4

√ 2⌈log2(n + 1)⌉ N

1 2⌈log2(n+1)⌉

.

June 4, 2013 Heraklion, Crete

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Classical Results Sn Reduction to dynamics Proofs General Case

Intrinsic Approximation on Sn

The goal: quantify the density of Sn ∩ Qn+1 in Sn (as opposed to approximating x ∈ Sn by rational points of Rn+1) Past results:

[Schmutz 2008] ∀ N > 1 ∀ x ∈ Sn ∃ p

q ∈ Sn with

q ≤ N and

x − p

q

< 4

√ 2⌈log2(n + 1)⌉ N

1 2⌈log2(n+1)⌉

.

June 4, 2013 Heraklion, Crete

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Classical Results Sn Reduction to dynamics Proofs General Case

Intrinsic Approximation on Sn

[Ghosh-Gorodnik-Nevo 2013] ∀ x ∈ Sn ∀ large enough N

∃ p

q ∈ Sn with

q ≤ N and

x − p

q

< 1

Nb for any      b < 1/4 n even b < 1/3 n = 3 b < 1

4 + 3 4n

n ≥ 5 odd

June 4, 2013 Heraklion, Crete

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Classical Results Sn Reduction to dynamics Proofs General Case

Intrinsic Approximation on Sn

[Ghosh-Gorodnik-Nevo 2013] ∀ x ∈ Sn ∀ large enough N

∃ p

q ∈ Sn with

q ≤ N and

x − p

q

< 1

Nb for any      b < 1/4 n even b < 1/3 n = 3 b < 1

4 + 3 4n

n ≥ 5 odd

June 4, 2013 Heraklion, Crete

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Classical Results Sn Reduction to dynamics Proofs General Case

Intrinsic Approximation on Sn

[Ghosh-Gorodnik-Nevo 2013] ∀ x ∈ Sn ∀ large enough N

∃ p

q ∈ Sn with

q ≤ N and

x − p

q

< 1

Nb for any      b < 1/4 n even b < 1/3 n = 3 b < 1

4 + 3 4n

n ≥ 5 odd

June 4, 2013 Heraklion, Crete

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Classical Results Sn Reduction to dynamics Proofs General Case

Intrinsic Approximation on Sn

[Ghosh-Gorodnik-Nevo 2013] ∀ x ∈ Sn ∀ large enough N

∃ p

q ∈ Sn with

q ≤ N and

x − p

q

< 1

Nb for any      b < 1/4 n even b < 1/3 n = 3 b < 1

4 + 3 4n

n ≥ 5 odd

June 4, 2013 Heraklion, Crete

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Classical Results Sn Reduction to dynamics Proofs General Case

Intrinsic Approximation on Sn

Theorem (Dirichlet for Sn) There exists a constant C such that for every x ∈ Sn and every N > 1, there exists p

q ∈ Sn with

q ≤ N and

x − p

q

<

C (qN)1/2 .

June 4, 2013 Heraklion, Crete

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Classical Results Sn Reduction to dynamics Proofs General Case

Intrinsic Approximation on Sn

Theorem (Dirichlet for Sn) There exists a constant C such that for every x ∈ Sn and every N > 1, there exists p

q ∈ Sn with

q ≤ N and

x − p

q

<

C (qN)1/2 .

June 4, 2013 Heraklion, Crete

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Classical Results Sn Reduction to dynamics Proofs General Case

Intrinsic Approximation on Sn

Corollary ∀ x ∈ Sn ∀ large enough N ∃ p

q ∈ Sn with

q ≤ N and

x − p

q

<

1 N1/2 . Corollary There exists C such that for every x ∈ Sn, ∃∞ p

q ∈ Sn such that

x − p

q

< C

q .

June 4, 2013 Heraklion, Crete

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Classical Results Sn Reduction to dynamics Proofs General Case

Intrinsic Approximation on Sn

Corollary ∀ x ∈ Sn ∀ large enough N ∃ p

q ∈ Sn with

q ≤ N and

x − p

q

<

1 N1/2 . Corollary There exists C such that for every x ∈ Sn, ∃∞ p

q ∈ Sn such that

x − p

q

< C

q .

June 4, 2013 Heraklion, Crete

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Classical Results Sn Reduction to dynamics Proofs General Case

Intrinsic Approximation on Sn

Corollary ∀ x ∈ Sn ∀ large enough N ∃ p

q ∈ Sn with

q ≤ N and

x − p

q

<

1 N1/2 . Corollary There exists C such that for every x ∈ Sn, ∃∞ p

q ∈ Sn such that

x − p

q

< C

q .

June 4, 2013 Heraklion, Crete

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Classical Results Sn Reduction to dynamics Proofs General Case

Intrinsic Approximation on Sn

Corollary ∀ x ∈ Sn ∀ large enough N ∃ p

q ∈ Sn with

q ≤ N and

x − p

q

<

1 N1/2 . Corollary There exists C such that for every x ∈ Sn, ∃∞ p

q ∈ Sn such that

x − p

q

< C

q .

June 4, 2013 Heraklion, Crete

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Classical Results Sn Reduction to dynamics Proofs General Case

Intrinsic Approximation on Sn

Define BA(Sn) :=

x ∈ Sn : ∃ c > 0 s.t.
x − p

q

> c

q ∀ p q ∈ Sn

.

Then similarly to results for Rd, we have Theorem (Jarník for Sn) BA(Sn) has full Hausdorff dimension in Sn. Moreover, BA(Sn) is absolutely winning for McMullen’s strengthening of Schmidt’s game.

June 4, 2013 Heraklion, Crete

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Classical Results Sn Reduction to dynamics Proofs General Case

Intrinsic Approximation on Sn

Define BA(Sn) :=

x ∈ Sn : ∃ c > 0 s.t.
x − p

q

> c

q ∀ p q ∈ Sn

.

Then similarly to results for Rd, we have Theorem (Jarník for Sn) BA(Sn) has full Hausdorff dimension in Sn. Moreover, BA(Sn) is absolutely winning for McMullen’s strengthening of Schmidt’s game.

June 4, 2013 Heraklion, Crete

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Classical Results Sn Reduction to dynamics Proofs General Case

Intrinsic Approximation on Sn

Define BA(Sn) :=

x ∈ Sn : ∃ c > 0 s.t.
x − p

q

> c

q ∀ p q ∈ Sn

.

Then similarly to results for Rd, we have Theorem (Jarník for Sn) BA(Sn) has full Hausdorff dimension in Sn. Moreover, BA(Sn) is absolutely winning for McMullen’s strengthening of Schmidt’s game.

June 4, 2013 Heraklion, Crete

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Classical Results Sn Reduction to dynamics Proofs General Case

Intrinsic Approximation on Sn

Define the set of ϕ-approximable points in Sn A(ϕ, Sn) :=

x ∈ Sn : ∃∞ p

q s.t.

x − p

q

< ϕ(q)
.

Theorem (Khintchine for Sn) Let ϕ : N → (0, ∞) such that k → kϕ(k) is decreasing. Then m

A(ϕ, Sn)
=
if

k kn−1ϕ(k)n < ∞,

1 if

k kn−1ϕ(k)n = ∞.

Here m denotes the normalized Lebesgue measure on Sn.

June 4, 2013 Heraklion, Crete

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Classical Results Sn Reduction to dynamics Proofs General Case

Intrinsic Approximation on Sn

Define the set of ϕ-approximable points in Sn A(ϕ, Sn) :=

x ∈ Sn : ∃∞ p

q s.t.

x − p

q

< ϕ(q)
.

Theorem (Khintchine for Sn) Let ϕ : N → (0, ∞) such that k → kϕ(k) is decreasing. Then m

A(ϕ, Sn)
=
if

k kn−1ϕ(k)n < ∞,

1 if

k kn−1ϕ(k)n = ∞.

Here m denotes the normalized Lebesgue measure on Sn.

June 4, 2013 Heraklion, Crete

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Classical Results Sn Reduction to dynamics Proofs General Case

Intrinsic Approximation on Sn

Define the set of ϕ-approximable points in Sn A(ϕ, Sn) :=

x ∈ Sn : ∃∞ p

q s.t.

x − p

q

< ϕ(q)
.

Theorem (Khintchine for Sn) Let ϕ : N → (0, ∞) such that k → kϕ(k) is decreasing. Then m

A(ϕ, Sn)
=
if

k kn−1ϕ(k)n < ∞,

1 if

k kn−1ϕ(k)n = ∞.

Here m denotes the normalized Lebesgue measure on Sn.

June 4, 2013 Heraklion, Crete

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Classical Results Sn Reduction to dynamics Proofs General Case

Reduction to dynamics

Let Q : Rn+2 → R be given by Q(y) =

n+1

i=1

y2

i − y2 n+2,

and let L :=

y ∈ Rn+2 : Q(y) = 0
⊂ Rn+2.

Then Sn ֒ → L, x → y = (x, 1) ∈ L.

June 4, 2013 Heraklion, Crete

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Classical Results Sn Reduction to dynamics Proofs General Case

Reduction to dynamics

Let Q : Rn+2 → R be given by Q(y) =

n+1

i=1

y2

i − y2 n+2,

and let L :=

y ∈ Rn+2 : Q(y) = 0
⊂ Rn+2.

Then Sn ֒ → L, x → y = (x, 1) ∈ L.

June 4, 2013 Heraklion, Crete

SLIDE 36

Classical Results Sn Reduction to dynamics Proofs General Case

Reduction to dynamics

Let Q : Rn+2 → R be given by Q(y) =

n+1

i=1

y2

i − y2 n+2,

and let L :=

y ∈ Rn+2 : Q(y) = 0
⊂ Rn+2.

Then Sn ֒ → L, x → y = (x, 1) ∈ L.

June 4, 2013 Heraklion, Crete

SLIDE 37

Classical Results Sn Reduction to dynamics Proofs General Case

Reduction to dynamics

Key Idea: p q ∈ Sn is close to x ∈ Sn ⇔ p q , 1

∈ L is close to (x, 1) ∈ L ⇔

(p, q) ∈ L ∩ Zn+2 is close to (qx, q) ∈ L. Therefore, we want to know how well the ‘lattice’ Λ0 := L ∩ Zn+2 approximates the line (x, 1) spanned by (x, 1).

June 4, 2013 Heraklion, Crete

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Classical Results Sn Reduction to dynamics Proofs General Case

Reduction to dynamics

Key Idea: p q ∈ Sn is close to x ∈ Sn ⇔ p q , 1

∈ L is close to (x, 1) ∈ L ⇔

(p, q) ∈ L ∩ Zn+2 is close to (qx, q) ∈ L. Therefore, we want to know how well the ‘lattice’ Λ0 := L ∩ Zn+2 approximates the line (x, 1) spanned by (x, 1).

June 4, 2013 Heraklion, Crete

SLIDE 39

Classical Results Sn Reduction to dynamics Proofs General Case

Reduction to dynamics

Key Idea: p q ∈ Sn is close to x ∈ Sn ⇔ p q , 1

∈ L is close to (x, 1) ∈ L ⇔

(p, q) ∈ L ∩ Zn+2 is close to (qx, q) ∈ L. Therefore, we want to know how well the ‘lattice’ Λ0 := L ∩ Zn+2 approximates the line (x, 1) spanned by (x, 1).

June 4, 2013 Heraklion, Crete

SLIDE 40

Classical Results Sn Reduction to dynamics Proofs General Case

Reduction to dynamics

Key Idea: p q ∈ Sn is close to x ∈ Sn ⇔ p q , 1

∈ L is close to (x, 1) ∈ L ⇔

(p, q) ∈ L ∩ Zn+2 is close to (qx, q) ∈ L. Therefore, we want to know how well the ‘lattice’ Λ0 := L ∩ Zn+2 approximates the line (x, 1) spanned by (x, 1).

June 4, 2013 Heraklion, Crete

SLIDE 41

Classical Results Sn Reduction to dynamics Proofs General Case

Reduction to dynamics

Key Idea (continued): If we contract (x, 1), close approximants to the line correspond to small vectors under the flow. Let rx ∈ K := SO(n + 1) send x to (1, 0, ..., 0) and (x, 1) to e1 := (1, 0, ..., 0, 1), and let gt ∈ G := SO(n + 1, 1) contract e1. Then good approximants to x correspond to small vectors in the lattice gtrxΛ0 for some time t > 0.

June 4, 2013 Heraklion, Crete

SLIDE 42

Classical Results Sn Reduction to dynamics Proofs General Case

Reduction to dynamics

Key Idea (continued): If we contract (x, 1), close approximants to the line correspond to small vectors under the flow. Let rx ∈ K := SO(n + 1) send x to (1, 0, ..., 0) and (x, 1) to e1 := (1, 0, ..., 0, 1), and let gt ∈ G := SO(n + 1, 1) contract e1. Then good approximants to x correspond to small vectors in the lattice gtrxΛ0 for some time t > 0.

June 4, 2013 Heraklion, Crete

SLIDE 43

Classical Results Sn Reduction to dynamics Proofs General Case

Reduction to dynamics

Key Idea (continued): If we contract (x, 1), close approximants to the line correspond to small vectors under the flow. Let rx ∈ K := SO(n + 1) send x to (1, 0, ..., 0) and (x, 1) to e1 := (1, 0, ..., 0, 1), and let gt ∈ G := SO(n + 1, 1) contract e1. Then good approximants to x correspond to small vectors in the lattice gtrxΛ0 for some time t > 0.

June 4, 2013 Heraklion, Crete

SLIDE 44

Classical Results Sn Reduction to dynamics Proofs General Case

Reduction to dynamics

Key Idea (continued): If we contract (x, 1), close approximants to the line correspond to small vectors under the flow. Let rx ∈ K := SO(n + 1) send x to (1, 0, ..., 0) and (x, 1) to e1 := (1, 0, ..., 0, 1), and let gt ∈ G := SO(n + 1, 1) contract e1. Then good approximants to x correspond to small vectors in the lattice gtrxΛ0 for some time t > 0.

June 4, 2013 Heraklion, Crete

SLIDE 45

Classical Results Sn Reduction to dynamics Proofs General Case

Reduction to dynamics

Key Idea (continued): If we contract (x, 1), close approximants to the line correspond to small vectors under the flow. Let rx ∈ K := SO(n + 1) send x to (1, 0, ..., 0) and (x, 1) to e1 := (1, 0, ..., 0, 1), and let gt ∈ G := SO(n + 1, 1) contract e1. Then good approximants to x correspond to small vectors in the lattice gtrxΛ0 for some time t > 0.

June 4, 2013 Heraklion, Crete

SLIDE 46

Classical Results Sn Reduction to dynamics Proofs General Case

Reduction to dynamics

Lemma (The correspondence)

1. Suppose q ≤ N and
x − p

q

<

ε

√

qN ≤ ε q ⇒

∃ t > 0 s.t. gtrx(p, q) < ε √ n + 1

q

N ≤ ε √ n + 1 .

2. Suppose gtrx(p, q) < δ and let N = etδ. Then

q ≤ N and

x − p

q

<

2δ

qN

.

June 4, 2013 Heraklion, Crete

SLIDE 47

Classical Results Sn Reduction to dynamics Proofs General Case

Reduction to dynamics

Lemma (The correspondence)

1. Suppose q ≤ N and
x − p

q

<

ε

√

qN ≤ ε q ⇒

∃ t > 0 s.t. gtrx(p, q) < ε √ n + 1

q

N ≤ ε √ n + 1 .

2. Suppose gtrx(p, q) < δ and let N = etδ. Then

q ≤ N and

x − p

q

<

2δ

qN

.

June 4, 2013 Heraklion, Crete

SLIDE 48

Classical Results Sn Reduction to dynamics Proofs General Case

Reduction to dynamics

Lemma (The correspondence)

1. Suppose q ≤ N and
x − p

q

<

ε

√

qN ≤ ε q ⇒

∃ t > 0 s.t. gtrx(p, q) < ε √ n + 1

q

N ≤ ε √ n + 1 .

2. Suppose gtrx(p, q) < δ and let N = etδ. Then

q ≤ N and

x − p

q

<

2δ

qN

.

June 4, 2013 Heraklion, Crete

SLIDE 49

Classical Results Sn Reduction to dynamics Proofs General Case

Reduction to dynamics

Lemma (The correspondence)

1. Suppose q ≤ N and
x − p

q

<

ε

√

qN ≤ ε q ⇒

∃ t > 0 s.t. gtrx(p, q) < ε √ n + 1

q

N ≤ ε √ n + 1 .

2. Suppose gtrx(p, q) < δ and let N = etδ. Then

q ≤ N and

x − p

q

<

2δ

qN

.

June 4, 2013 Heraklion, Crete

SLIDE 50

Classical Results Sn Reduction to dynamics Proofs General Case

Reduction to dynamics

Lemma (The correspondence)

1. Suppose q ≤ N and
x − p

q

<

ε

√

qN ≤ ε q ⇒

∃ t > 0 s.t. gtrx(p, q) < ε √ n + 1

q

N ≤ ε √ n + 1 .

2. Suppose gtrx(p, q) < δ and let N = etδ. Then

q ≤ N and

x − p

q

<

2δ

qN

.

June 4, 2013 Heraklion, Crete

SLIDE 51

Classical Results Sn Reduction to dynamics Proofs General Case

Reduction to dynamics

Proof of 2. gtrx(p, q) < δ , N = etδ ⇒    q ≤ N

x − p

q

<

2δ

√

qN

Since e1 = (1, 0, ..., 0, 1) is the contracting eigenvector for gt,

ne has

Ne−t = δ > gtrx(p, q) ≥ gtqe1 = qe−t ⇒ q ≤ N.

June 4, 2013 Heraklion, Crete

SLIDE 52

Classical Results Sn Reduction to dynamics Proofs General Case

Reduction to dynamics

Proof of 2. gtrx(p, q) < δ , N = etδ ⇒    q ≤ N

x − p

q

<

2δ

√

qN

Since e1 = (1, 0, ..., 0, 1) is the contracting eigenvector for gt,

ne has

Ne−t = δ > gtrx(p, q) ≥ gtqe1 = qe−t ⇒ q ≤ N.

June 4, 2013 Heraklion, Crete

SLIDE 53

Classical Results Sn Reduction to dynamics Proofs General Case

Reduction to dynamics

Proof of 2. gtrx(p, q) < δ , N = etδ ⇒    q ≤ N

x − p

q

<

2δ

√

qN

Since e1 = (1, 0, ..., 0, 1) is the contracting eigenvector for gt,

ne has

Ne−t = δ > gtrx(p, q) ≥ gtqe1 = qe−t ⇒ q ≤ N.

June 4, 2013 Heraklion, Crete

SLIDE 54

Classical Results Sn Reduction to dynamics Proofs General Case

Reduction to dynamics

Proof of 2. gtrx(p, q) < δ , N = etδ ⇒    q ≤ N

x − p

q

<

2δ

√

qN

Since e1 = (1, 0, ..., 0, 1) is the contracting eigenvector for gt,

ne has

Ne−t = δ > gtrx(p, q) ≥ gtqe1 = qe−t ⇒ q ≤ N.

June 4, 2013 Heraklion, Crete

SLIDE 55

Classical Results Sn Reduction to dynamics Proofs General Case

Reduction to dynamics

Proof of 2. gtrx(p, q) < δ , N = etδ ⇒    q ≤ N

x − p

q

<

2δ

√

qN

Since e1 = (1, 0, ..., 0, 1) is the contracting eigenvector for gt,

ne has

Ne−t = δ > gtrx(p, q) ≥ gtqe1 = qe−t ⇒ q ≤ N.

June 4, 2013 Heraklion, Crete

SLIDE 56

Classical Results Sn Reduction to dynamics Proofs General Case

Reduction to dynamics

Proof of 2. gtrx(p, q) < δ , N = etδ ⇒    q ≤ N

x − p

q

<

2δ

√

qN

Since e1 = (1, 0, ..., 0, 1) is the contracting eigenvector for gt,

ne has

Ne−t = δ > gtrx(p, q) ≥ gtqe1 = qe−t ⇒ q ≤ N.

June 4, 2013 Heraklion, Crete

SLIDE 57

Classical Results Sn Reduction to dynamics Proofs General Case

Reduction to dynamics

Proof of 2. gtrx(p, q) < δ , N = etδ ⇒    q ≤ N

x − p

q

<

2δ

√

qN

Since e1 = (1, 0, ..., 0, 1) is the contracting eigenvector for gt,

ne has

Ne−t = δ > gtrx(p, q) ≥ gtqe1 = qe−t ⇒ q ≤ N.

June 4, 2013 Heraklion, Crete

SLIDE 58

Classical Results Sn Reduction to dynamics Proofs General Case

Reduction to dynamics

Proof of 2. gtrx(p, q) < δ , N = etδ ⇒    q ≤ N

x − p

q

<

2δ

√

qN

x − p

q

≤
x − p

q

e

=

2
1 − x · p

q

=
2
1 − rx

p q , 1

=

1 √q

2
q − rx(p, q)0
=

2

qet
1

√ 2 1 √ 2

q − rx(p, q)0
et

= 2

qet
1

√ 2 gtrx(p, q)e < 2 √ δ

qet =

2δ

qN

.

June 4, 2013 Heraklion, Crete

SLIDE 59

Classical Results Sn Reduction to dynamics Proofs General Case

Reduction to dynamics

Proof of 2. gtrx(p, q) < δ , N = etδ ⇒    q ≤ N

x − p

q

<

2δ

√

qN

x − p

q

≤
x − p

q

e

=

2
1 − x · p

q

=
2
1 − rx

p q , 1

=

1 √q

2
q − rx(p, q)0
=

2

qet
1

√ 2 1 √ 2

q − rx(p, q)0
et

= 2

qet
1

√ 2 gtrx(p, q)e < 2 √ δ

qet =

2δ

qN

.

June 4, 2013 Heraklion, Crete

SLIDE 60

Classical Results Sn Reduction to dynamics Proofs General Case

Reduction to dynamics

Proof of 2. gtrx(p, q) < δ , N = etδ ⇒    q ≤ N

x − p

q

<

2δ

√

qN

x − p

q

≤
x − p

q

e

=

2
1 − x · p

q

=
2
1 − rx

p q , 1

=

1 √q

2
q − rx(p, q)0
=

2

qet
1

√ 2 1 √ 2

q − rx(p, q)0
et

= 2

qet
1

√ 2 gtrx(p, q)e < 2 √ δ

qet =

2δ

qN

.

June 4, 2013 Heraklion, Crete

SLIDE 61

Classical Results Sn Reduction to dynamics Proofs General Case

Reduction to dynamics

Proof of 2. gtrx(p, q) < δ , N = etδ ⇒    q ≤ N

x − p

q

<

2δ

√

qN

x − p

q

≤
x − p

q

e

=

2
1 − x · p

q

=
2
1 − rx

p q , 1

=

1 √q

2
q − rx(p, q)0
=

2

qet
1

√ 2 1 √ 2

q − rx(p, q)0
et

= 2

qet
1

√ 2 gtrx(p, q)e < 2 √ δ

qet =

2δ

qN

.

June 4, 2013 Heraklion, Crete

SLIDE 62

Classical Results Sn Reduction to dynamics Proofs General Case

Reduction to dynamics

Proof of 2. gtrx(p, q) < δ , N = etδ ⇒    q ≤ N

x − p

q

<

2δ

√

qN

x − p

q

≤
x − p

q

e

=

2
1 − x · p

q

=
2
1 − rx

p q , 1

=

1 √q

2
q − rx(p, q)0
=

2

qet
1

√ 2 1 √ 2

q − rx(p, q)0
et

= 2

qet
1

√ 2 gtrx(p, q)e < 2 √ δ

qet =

2δ

qN

.

June 4, 2013 Heraklion, Crete

SLIDE 63

Classical Results Sn Reduction to dynamics Proofs General Case

Reduction to dynamics

Proof of 2. gtrx(p, q) < δ , N = etδ ⇒    q ≤ N

x − p

q

<

2δ

√

qN

x − p

q

≤
x − p

q

e

=

2
1 − x · p

q

=
2
1 − rx

p q , 1

=

1 √q

2
q − rx(p, q)0
=

2

qet
1

√ 2 1 √ 2

q − rx(p, q)0
et

= 2

qet
1

√ 2 gtrx(p, q)e < 2 √ δ

qet =

2δ

qN

.

June 4, 2013 Heraklion, Crete

SLIDE 64

Classical Results Sn Reduction to dynamics Proofs General Case

Reduction to dynamics

Proof of 2. gtrx(p, q) < δ , N = etδ ⇒    q ≤ N

x − p

q

<

2δ

√

qN

x − p

q

≤
x − p

q

e

=

2
1 − x · p

q

=
2
1 − rx

p q , 1

=

1 √q

2
q − rx(p, q)0
=

2

qet
1

√ 2 1 √ 2

q − rx(p, q)0
et

= 2

qet
1

√ 2 gtrx(p, q)e < 2 √ δ

qet =

2δ

qN

.

June 4, 2013 Heraklion, Crete

SLIDE 65

Classical Results Sn Reduction to dynamics Proofs General Case

Reduction to dynamics

Proof of 2. gtrx(p, q) < δ , N = etδ ⇒    q ≤ N

x − p

q

<

2δ

√

qN

x − p

q

≤
x − p

q

e

=

2
1 − x · p

q

=
2
1 − rx

p q , 1

=

1 √q

2
q − rx(p, q)0
=

2

qet
1

√ 2 1 √ 2

q − rx(p, q)0
et

= 2

qet
1

√ 2 gtrx(p, q)e < 2 √ δ

qet =

2δ

qN

.

June 4, 2013 Heraklion, Crete

SLIDE 66

Classical Results Sn Reduction to dynamics Proofs General Case

Reduction to dynamics

Proof of 2. gtrx(p, q) < δ , N = etδ ⇒    q ≤ N

x − p

q

<

2δ

√

qN

x − p

q

≤
x − p

q

e

=

2
1 − x · p

q

=
2
1 − rx

p q , 1

=

1 √q

2
q − rx(p, q)0
=

2

qet
1

√ 2 1 √ 2

q − rx(p, q)0
et

= 2

qet
1

√ 2 gtrx(p, q)e < 2 √ δ

qet =

2δ

qN

.

June 4, 2013 Heraklion, Crete

SLIDE 67

Classical Results Sn Reduction to dynamics Proofs General Case

Proofs

Thus we have reduced the problem of approximation by rational points on Sn to gt-dynamics on the space L := GΛ0 = G/Γ, where Γ := {γ ∈ G : γΛ0 = Λ0}, a lattice in G containing GZ as a subgroup of finite index. For Λ ∈ L, define ω(Λ) := min

v∈Λ0 v .

Proposition (Mahler’s Compactness) For any ε > 0, the set {Λ ∈ L : ω(Λ) ≥ ε} is compact.

June 4, 2013 Heraklion, Crete

SLIDE 68

Classical Results Sn Reduction to dynamics Proofs General Case

Proofs

Thus we have reduced the problem of approximation by rational points on Sn to gt-dynamics on the space L := GΛ0 = G/Γ, where Γ := {γ ∈ G : γΛ0 = Λ0}, a lattice in G containing GZ as a subgroup of finite index. For Λ ∈ L, define ω(Λ) := min

v∈Λ0 v .

Proposition (Mahler’s Compactness) For any ε > 0, the set {Λ ∈ L : ω(Λ) ≥ ε} is compact.

June 4, 2013 Heraklion, Crete

SLIDE 69

Classical Results Sn Reduction to dynamics Proofs General Case

Proofs

Thus we have reduced the problem of approximation by rational points on Sn to gt-dynamics on the space L := GΛ0 = G/Γ, where Γ := {γ ∈ G : γΛ0 = Λ0}, a lattice in G containing GZ as a subgroup of finite index. For Λ ∈ L, define ω(Λ) := min

v∈Λ0 v .

Proposition (Mahler’s Compactness) For any ε > 0, the set {Λ ∈ L : ω(Λ) ≥ ε} is compact.

June 4, 2013 Heraklion, Crete

SLIDE 70

Classical Results Sn Reduction to dynamics Proofs General Case

Proofs

Thus we have reduced the problem of approximation by rational points on Sn to gt-dynamics on the space L := GΛ0 = G/Γ, where Γ := {γ ∈ G : γΛ0 = Λ0}, a lattice in G containing GZ as a subgroup of finite index. For Λ ∈ L, define ω(Λ) := min

v∈Λ0 v .

Proposition (Mahler’s Compactness) For any ε > 0, the set {Λ ∈ L : ω(Λ) ≥ ε} is compact.

June 4, 2013 Heraklion, Crete

SLIDE 71

Classical Results Sn Reduction to dynamics Proofs General Case

Proofs

Corollary (Minkowski’s Lemma) ∃ C > 0 such that ω(Λ) < C ∀ Λ ∈ L. Proof. If not, have a sequence Λk ∈ L with ω(Λk) → ∞, which cannot have a limit point, contradicting to Mahler’s Criterion.

June 4, 2013 Heraklion, Crete

SLIDE 72

Classical Results Sn Reduction to dynamics Proofs General Case

Proofs

Corollary (Minkowski’s Lemma) ∃ C > 0 such that ω(Λ) < C ∀ Λ ∈ L. Proof. If not, have a sequence Λk ∈ L with ω(Λk) → ∞, which cannot have a limit point, contradicting to Mahler’s Criterion.

June 4, 2013 Heraklion, Crete

SLIDE 73

Classical Results Sn Reduction to dynamics Proofs General Case

Proofs

Corollary (Minkowski’s Lemma) ∃ C > 0 such that ω(Λ) < C ∀ Λ ∈ L. Proof. If not, have a sequence Λk ∈ L with ω(Λk) → ∞, which cannot have a limit point, contradicting to Mahler’s Criterion.

June 4, 2013 Heraklion, Crete

SLIDE 74

Classical Results Sn Reduction to dynamics Proofs General Case

Proofs

Proof (Dirichlet for Sn). Take C as above, let N > C, and choose t such that N = etC. Then by Minkowski, ∃ (p, q) such that gtrx(p, q) < C. Hence, by part 2 of the Correspondence Lemma, we have q ≤ N and

x − p

q

<

2C

qN

. And if N ≤ C, then

x − p

1

< 2 ≤

2C √ 1 · N .

June 4, 2013 Heraklion, Crete

SLIDE 75

Classical Results Sn Reduction to dynamics Proofs General Case

Proofs

Proof (Dirichlet for Sn). Take C as above, let N > C, and choose t such that N = etC. Then by Minkowski, ∃ (p, q) such that gtrx(p, q) < C. Hence, by part 2 of the Correspondence Lemma, we have q ≤ N and

x − p

q

<

2C

qN

. And if N ≤ C, then

x − p

1

< 2 ≤

2C √ 1 · N .

June 4, 2013 Heraklion, Crete

SLIDE 76

Classical Results Sn Reduction to dynamics Proofs General Case

Proofs

Proof (Dirichlet for Sn). Take C as above, let N > C, and choose t such that N = etC. Then by Minkowski, ∃ (p, q) such that gtrx(p, q) < C. Hence, by part 2 of the Correspondence Lemma, we have q ≤ N and

x − p

q

<

2C

qN

. And if N ≤ C, then

x − p

1

< 2 ≤

2C √ 1 · N .

June 4, 2013 Heraklion, Crete

SLIDE 77

Classical Results Sn Reduction to dynamics Proofs General Case

Proofs

Proof (Dirichlet for Sn). Take C as above, let N > C, and choose t such that N = etC. Then by Minkowski, ∃ (p, q) such that gtrx(p, q) < C. Hence, by part 2 of the Correspondence Lemma, we have q ≤ N and

x − p

q

<

2C

qN

. And if N ≤ C, then

x − p

1

< 2 ≤

2C √ 1 · N .

June 4, 2013 Heraklion, Crete

SLIDE 78

Classical Results Sn Reduction to dynamics Proofs General Case

Proofs

Proof (Jarník for Sn). From the Correspondence Lemma it follows that x ∈ BA(Sn) if and only if the trajectory {gtrxΛ0 : t > 0} is bounded in L . Thus full Hausdorff dimension of BA(Sn) follows from Dani’s work on bounded trajectories, and the absolute winning property – from the work of McMullen.

June 4, 2013 Heraklion, Crete

SLIDE 79

Classical Results Sn Reduction to dynamics Proofs General Case

Proofs

Proof (Jarník for Sn). From the Correspondence Lemma it follows that x ∈ BA(Sn) if and only if the trajectory {gtrxΛ0 : t > 0} is bounded in L . Thus full Hausdorff dimension of BA(Sn) follows from Dani’s work on bounded trajectories, and the absolute winning property – from the work of McMullen.

June 4, 2013 Heraklion, Crete

SLIDE 80

Classical Results Sn Reduction to dynamics Proofs General Case

Proofs

Proof (Jarník for Sn). From the Correspondence Lemma it follows that x ∈ BA(Sn) if and only if the trajectory {gtrxΛ0 : t > 0} is bounded in L . Thus full Hausdorff dimension of BA(Sn) follows from Dani’s work on bounded trajectories, and the absolute winning property – from the work of McMullen.

June 4, 2013 Heraklion, Crete

SLIDE 81

Classical Results Sn Reduction to dynamics Proofs General Case

Proofs

Proof (Jarník for Sn). From the Correspondence Lemma it follows that x ∈ BA(Sn) if and only if the trajectory {gtrxΛ0 : t > 0} is bounded in L . Thus full Hausdorff dimension of BA(Sn) follows from Dani’s work on bounded trajectories, and the absolute winning property – from the work of McMullen.

June 4, 2013 Heraklion, Crete

SLIDE 82

Classical Results Sn Reduction to dynamics Proofs General Case

Proofs

Proof (Khintchine for Sn). The Correspondence Lemma translates the statement of the theorem to a dynamical Borel-Cantelli lemma for the gt-action

n L.

The latter can be derived from the work of Kleinbock-Margulis

n logarithm laws for flows on homogeneous spaces.

Note: via the Mass Transference Principle due to Beresnevich and Velani, the 0-1 law for Lebesgue measure implies a corresponding statement for Hausdofff measures.

June 4, 2013 Heraklion, Crete

SLIDE 83

Classical Results Sn Reduction to dynamics Proofs General Case

Proofs

Proof (Khintchine for Sn). The Correspondence Lemma translates the statement of the theorem to a dynamical Borel-Cantelli lemma for the gt-action

n L.

The latter can be derived from the work of Kleinbock-Margulis

n logarithm laws for flows on homogeneous spaces.

Note: via the Mass Transference Principle due to Beresnevich and Velani, the 0-1 law for Lebesgue measure implies a corresponding statement for Hausdofff measures.

June 4, 2013 Heraklion, Crete

SLIDE 84

Classical Results Sn Reduction to dynamics Proofs General Case

Proofs

Proof (Khintchine for Sn). The Correspondence Lemma translates the statement of the theorem to a dynamical Borel-Cantelli lemma for the gt-action

n L.

The latter can be derived from the work of Kleinbock-Margulis

n logarithm laws for flows on homogeneous spaces.

Note: via the Mass Transference Principle due to Beresnevich and Velani, the 0-1 law for Lebesgue measure implies a corresponding statement for Hausdofff measures.

June 4, 2013 Heraklion, Crete

SLIDE 85

Classical Results Sn Reduction to dynamics Proofs General Case

P .S.

A generalization to arbitrary quadratic varieties containing a dense set of rational points is work in progress, joint with

L. Fishman and D. Simmons

June 4, 2013 Heraklion, Crete