S n Classical Results 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
S n Classical Results Reduction to dynamics Proofs General Case Motivation and Classical Results Theorem (Dirichlet’s Theorem on Simultaneous Approximation) For any N > 1 and every x ∈ R d , there exists p ∈ Z d and q ∈ N such that � � � x − p 1 � � q ≤ N and � < qN 1 / d . � � q June 4, 2013 Heraklion, Crete
S n Classical Results Reduction to dynamics Proofs General Case Motivation and Classical Results Theorem (Dirichlet’s Theorem on Simultaneous Approximation) For any N > 1 and every x ∈ R d , there exists p ∈ Z d and q ∈ N such that � � � x − p 1 � � q ≤ N and � < qN 1 / d . � � q June 4, 2013 Heraklion, Crete
S n Classical Results Reduction to dynamics Proofs General Case Motivation and Classical Results Theorem (Dirichlet’s Theorem on Simultaneous Approximation) For any N > 1 and every x ∈ R d , there exists p ∈ Z d and q ∈ N such that � � � x − p 1 � � q ≤ N and � < qN 1 / d . � � q June 4, 2013 Heraklion, Crete
S n Classical Results Reduction to dynamics Proofs General Case Motivation and Classical Results Theorem (Dirichlet’s Theorem on Simultaneous Approximation) For any N > 1 and every x ∈ R d , there exists p ∈ Z d and q ∈ N such that � � � x − p 1 � � q ≤ N and � < qN 1 / d . � � q June 4, 2013 Heraklion, Crete
S n Classical Results Reduction to dynamics Proofs General Case Motivation and Classical Results Corollary For every x ∈ R d ∃ ∞ p q ∈ R d such that � � � x − p 1 � � � < q 1 + 1 / d . � � q June 4, 2013 Heraklion, Crete
S n Classical Results Reduction to dynamics Proofs General Case Motivation and Classical Results Corollary For every x ∈ R d ∃ ∞ p q ∈ R d such that � � � x − p 1 � � � < q 1 + 1 / d . � � q June 4, 2013 Heraklion, Crete
S n Classical Results Reduction to dynamics Proofs General Case Motivation and Classical Results Define � � � x − p � c ∀ p � x ∈ R d : ∃ c > 0 s.t. q ∈ R d � � BA := � > . � � q q 1 + 1 / d Theorem (Jarník) BA has full Hausdorff dimension. In fact even stronger, it is winning for Schmidt’s game. June 4, 2013 Heraklion, Crete
S n Classical Results Reduction to dynamics Proofs General Case Motivation and Classical Results Define � � � x − p � c ∀ p � x ∈ R d : ∃ c > 0 s.t. q ∈ R d � � BA := � > . � � q q 1 + 1 / d Theorem (Jarník) BA has full Hausdorff dimension. In fact even stronger, it is winning for Schmidt’s game. June 4, 2013 Heraklion, Crete
S n Classical Results Reduction to dynamics Proofs General Case Motivation and Classical Results Define � � � x − p � c ∀ p � x ∈ R d : ∃ c > 0 s.t. q ∈ R d � � BA := � > . � � q q 1 + 1 / d Theorem (Jarník) BA has full Hausdorff dimension. In fact even stronger, it is winning for Schmidt’s game. June 4, 2013 Heraklion, Crete
S n Classical Results Reduction to dynamics Proofs General Case Motivation and Classical Results For ϕ : N → ( 0 , ∞ ) , we define the set of ϕ -approximable points � x ∈ R d : ∃ ∞ p � x − p � � � � � A ( ϕ ) := q s.t. � < ϕ ( q ) . � � q Theorem (Khintchine) Let ϕ : N → ( 0 , ∞ ) such that k �→ k ϕ ( k ) is non-increasing. Then k k d ϕ ( k ) d < ∞ , � 0 if � � � m A ( ϕ ) = k k d ϕ ( k ) d = ∞ . ∞ if � Here m denotes Lebesgue measure. June 4, 2013 Heraklion, Crete
S n Classical Results Reduction to dynamics Proofs General Case Motivation and Classical Results For ϕ : N → ( 0 , ∞ ) , we define the set of ϕ -approximable points � x ∈ R d : ∃ ∞ p � x − p � � � � � A ( ϕ ) := q s.t. � < ϕ ( q ) . � � q Theorem (Khintchine) Let ϕ : N → ( 0 , ∞ ) such that k �→ k ϕ ( k ) is non-increasing. Then k k d ϕ ( k ) d < ∞ , � 0 if � � � m A ( ϕ ) = k k d ϕ ( k ) d = ∞ . ∞ if � Here m denotes Lebesgue measure. June 4, 2013 Heraklion, Crete
S n Classical Results Reduction to dynamics Proofs General Case Motivation and Classical Results For ϕ : N → ( 0 , ∞ ) , we define the set of ϕ -approximable points � x ∈ R d : ∃ ∞ p � x − p � � � � � A ( ϕ ) := q s.t. � < ϕ ( q ) . � � q Theorem (Khintchine) Let ϕ : N → ( 0 , ∞ ) such that k �→ k ϕ ( k ) is non-increasing. Then k k d ϕ ( k ) d < ∞ , � 0 if � � � m A ( ϕ ) = k k d ϕ ( k ) d = ∞ . ∞ if � Here m denotes Lebesgue measure. June 4, 2013 Heraklion, Crete
S n Classical Results Reduction to dynamics Proofs General Case Intrinsic Approximation on S n The goal: quantify the density of S n ∩ Q n + 1 in S n (as opposed to approximating x ∈ S n by rational points of R n + 1 ) Past results: • [Schmutz 2008] ∀ N > 1 ∀ x ∈ S n ∃ p q ∈ S n with √ � � � x − p � < 4 2 ⌈ log 2 ( n + 1 ) ⌉ � � q ≤ N and . � � 1 q N 2 ⌈ log2 ( n + 1 ) ⌉ June 4, 2013 Heraklion, Crete
S n Classical Results Reduction to dynamics Proofs General Case Intrinsic Approximation on S n The goal: quantify the density of S n ∩ Q n + 1 in S n (as opposed to approximating x ∈ S n by rational points of R n + 1 ) Past results: • [Schmutz 2008] ∀ N > 1 ∀ x ∈ S n ∃ p q ∈ S n with √ � � � x − p � < 4 2 ⌈ log 2 ( n + 1 ) ⌉ � � q ≤ N and . � � 1 q N 2 ⌈ log2 ( n + 1 ) ⌉ June 4, 2013 Heraklion, Crete
S n Classical Results Reduction to dynamics Proofs General Case Intrinsic Approximation on S n The goal: quantify the density of S n ∩ Q n + 1 in S n (as opposed to approximating x ∈ S n by rational points of R n + 1 ) Past results: • [Schmutz 2008] ∀ N > 1 ∀ x ∈ S n ∃ p q ∈ S n with √ � � � x − p � < 4 2 ⌈ log 2 ( n + 1 ) ⌉ � � q ≤ N and . � � 1 q N 2 ⌈ log2 ( n + 1 ) ⌉ June 4, 2013 Heraklion, Crete
S n Classical Results Reduction to dynamics Proofs General Case Intrinsic Approximation on S n The goal: quantify the density of S n ∩ Q n + 1 in S n (as opposed to approximating x ∈ S n by rational points of R n + 1 ) Past results: • [Schmutz 2008] ∀ N > 1 ∀ x ∈ S n ∃ p q ∈ S n with √ � � � x − p � < 4 2 ⌈ log 2 ( n + 1 ) ⌉ � � q ≤ N and . � � 1 q N 2 ⌈ log2 ( n + 1 ) ⌉ June 4, 2013 Heraklion, Crete
S n Classical Results Reduction to dynamics Proofs General Case Intrinsic Approximation on S n • [Ghosh-Gorodnik-Nevo 2013] ∀ x ∈ S n ∀ large enough N q ∈ S n with ∃ p � � � < 1 � x − p � � q ≤ N and � � N b q for any b < 1 / 4 n even b < 1 / 3 n = 3 b < 1 4 + 3 n ≥ 5 odd 4 n June 4, 2013 Heraklion, Crete
S n Classical Results Reduction to dynamics Proofs General Case Intrinsic Approximation on S n • [Ghosh-Gorodnik-Nevo 2013] ∀ x ∈ S n ∀ large enough N q ∈ S n with ∃ p � � � < 1 � x − p � � q ≤ N and � � N b q for any b < 1 / 4 n even b < 1 / 3 n = 3 b < 1 4 + 3 n ≥ 5 odd 4 n June 4, 2013 Heraklion, Crete
S n Classical Results Reduction to dynamics Proofs General Case Intrinsic Approximation on S n • [Ghosh-Gorodnik-Nevo 2013] ∀ x ∈ S n ∀ large enough N q ∈ S n with ∃ p � � � < 1 � x − p � � q ≤ N and � � N b q for any b < 1 / 4 n even b < 1 / 3 n = 3 b < 1 4 + 3 n ≥ 5 odd 4 n June 4, 2013 Heraklion, Crete
S n Classical Results Reduction to dynamics Proofs General Case Intrinsic Approximation on S n • [Ghosh-Gorodnik-Nevo 2013] ∀ x ∈ S n ∀ large enough N q ∈ S n with ∃ p � � � < 1 � x − p � � q ≤ N and � � N b q for any b < 1 / 4 n even b < 1 / 3 n = 3 b < 1 4 + 3 n ≥ 5 odd 4 n June 4, 2013 Heraklion, Crete
S n Classical Results Reduction to dynamics Proofs General Case Intrinsic Approximation on S n Theorem (Dirichlet for S n ) There exists a constant C such that for every x ∈ S n and every q ∈ S n with N > 1 , there exists p � � � x − p C � � q ≤ N and � < ( qN ) 1 / 2 . � � q June 4, 2013 Heraklion, Crete
S n Classical Results Reduction to dynamics Proofs General Case Intrinsic Approximation on S n Theorem (Dirichlet for S n ) There exists a constant C such that for every x ∈ S n and every q ∈ S n with N > 1 , there exists p � � � x − p C � � q ≤ N and � < ( qN ) 1 / 2 . � � q June 4, 2013 Heraklion, Crete
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