Hausdorff dimension of weighted singular vectors in R 2

Transcription

1 Hausdorff dimension of weighted singular vectors in R 2 Lingmin LIAO (joint with Ronggang Shi, Omri N. Solan, and Nattalie Tamam) Université Paris-Est NCTS Workshop on Dynamical Systems August 15th 2016 Lingmin LIAO, Université Paris-Est Hausdorff dimension of weighted singular vectors in R 2 1/28

2 Outline 1 Uniform Diophantine approximation 2 Singular vectors and uniform approximation 3 Lower bound 4 Upper bound Lingmin LIAO, Université Paris-Est Hausdorff dimension of weighted singular vectors in R 2 2/28

3 I. Dirichlet and Legendre Denote by the distance to the nearest integer. Dirichlet Theorem, 1842 (uniform approximation) : Let θ, Q be real numbers with Q 1. There exists an integer n with 1 n Q, such that nθ < Q 1. In other words, { θ : Q 1, nθ < Q 1 has a solution 1 n Q } = R. Corollary (asymptotic approximation) : For any real θ, there exist infinitely many integers n such that nθ < n 1. In other words, { θ : nθ < n 1 for infinitely many n } = R. Legendre 1808 Essai sur la théorie des nombres : proved the asymptotic approximation property by using continued fractions. Lingmin LIAO, Université Paris-Est Hausdorff dimension of weighted singular vectors in R 2 3/28

4 II. Approximation with a higher speed Jarník 1929, Besicovith 1934 : For w > 1, the Hausdorff dimension dim H (L w ) = dim H { θ : nθ < n w infinitely many n } = 2 w + 1. What is about the set U w := { θ : Q > 1, nθ < Q w has a solution 1 n Q }? Khintchine 1926 : For w > 1, U w is empty. Proof : Apply the continued fraction theory. Lingmin LIAO, Université Paris-Est Hausdorff dimension of weighted singular vectors in R 2 4/28

5 III. Inhomogeneous approximation Bugeaud 2003, Troubetzkoy Schmeling 2003 : for all θ R \ Q, w 1, set L w [θ] :={y : nθ y < n w for infinitely many n}. Then dim H (L w [θ]) = 1/w. Question of Bugeaud Laurent 2005 : for a fixed irrational θ, what is the size (Hausdorff dimension) of the set U w [θ] := { y : Q 1, nθ y < Q w has a solution 1 n Q }. Remark : U w [θ] \ {nθ : n N} L w [θ]. Lingmin LIAO, Université Paris-Est Hausdorff dimension of weighted singular vectors in R 2 5/28

6 IV. Inhomogeneous uniform approximation For θ Q, define w(θ) := sup{β > 0 : lim inf j j β jθ = 0} 1. D.H. Kim L, arxiv 2015 : 1 2 w w w 0 w 0 1/w case 1 : w 0 = w(θ) > case 2 : w(θ) = 1 1/w Remark : for Liouville numbers (w(θ) = ), dim H (U w [θ]) = 0 w 1. Lingmin LIAO, Université Paris-Est Hausdorff dimension of weighted singular vectors in R 2 6/28

7 V. Diophantine approximation of β-transformation Let β > 1 be a real number and T β be the β-transformation defined by Shen Wang 2013 : for x [0, 1], T β x = βx mod 1. dim H {x [0, 1] : T n β (x) < (β n ) v, for infinitely many n} = v. Persson Schmeling 2008 : dim H {β > 1 : T n β (1) < (β n ) v, for infinitely many n} = v. Bugeaud L, 2016 : ( 1 v ) 2, dim H {x : N 1, 1 n N, Tβ n (x) < (β N ) v } = 1 + v ( 1 v ) 2. dim H {β > 1 : N 1, 1 n N, Tβ n (1) < (β N ) v } = 1 + v Lingmin LIAO, Université Paris-Est Hausdorff dimension of weighted singular vectors in R 2 7/28

8 VI. Relation with the hitting time Let (T θ ) θ Θ (Θ R) be a family of systems on a metric space (X, d). Define τr θ (x, y) = inf{n : Tθ n x B(y, r)}. and define (for the zero entropy systems) R θ (x, y) := lim inf r 0 We have (fixing x, y X) log τr θ (x, y), R θ (x, y) := lim sup log r r 0 log τr θ (x, y). log r L w = {θ : d(t n θ x, y) < n w for infinitely many n} {θ : R θ (x, y) 1/w}, U w = {θ : N 1, 1 n N, d(t n θ x, y) < N w } {θ : R θ (x, y) 1/w}. Thus, U w is (almost) less than L w. The same thing holds when fixing (θ, x) or (θ, y). Positive entropy systems, sometimes : replace log τ r (x, y) by τ r (x, y). Lingmin LIAO, Université Paris-Est Hausdorff dimension of weighted singular vectors in R 2 8/28

9 VII. Level sets of hitting time Shrinking target problem : Fix one dynamical system T, fix one point y, one studies the size of the level set {x X : R(x, y) = α}, for a given α. Measure results : Boshernitzan, Chernov, Chazottes, Fayad, Galatalo, Kleinbock, Kim... Hausdorff dimension results : Hill Velani 1995, 1999 ; Urbański 2002 ; Fernández Melián Pestana 2007 ; Shen Wang 2013 ; Li Wang Wu Xu 2014, Bugeaud Wang For sets of parameters : Persson-Schmeling 2008, Li-Persson-Wang-Wu 2014, Aspenberg-Persson Dynamical diophantine approximation problem : Fix one dynamical system T, fix one point x, one studies the size of the level set {y X : R(x, y) = α}, for a given α. Fan Schmeling-Troubetzkoy 2013 ; Liao Seuret 2013 ; Persson Rams Lingmin LIAO, Université Paris-Est Hausdorff dimension of weighted singular vectors in R 2 9/28

10 Singular vectors and uniform approximation Lingmin LIAO, Université Paris-Est Hausdorff dimension of weighted singular vectors in R 2 10/28

11 I. Dirichlet Minkowski A special case of Minkowski s linear form theorem gives the following Dirichlet s theorem with weight w : Let w = (w 1, w 2 ) be a pair of positive real numbers such that w 1 + w 2 = 1. Then for all x = (x 1, x 2 ) R 2 and T > 1 there is (p, q) = (p 1, p 2, q) Z 2 Z such that qx 1 p 1 < T w1 qx 2 p 2 < T w2. 0 < q T A vector x = (x 1, x 2 ) R 2 is said to be w-singular if for every ε > 0 there exists T 0 > 1 such that for all T > T 0 the system of inequalities qx 1 p 1 < ε w1 T w1 qx 2 p 2 < ε w2 T w2 (1) 0 < q T admits an integer solution (p, q) Z 2 Z. Lingmin LIAO, Université Paris-Est Hausdorff dimension of weighted singular vectors in R 2 11/28

12 II. ε-improvable Dirichlet theorem For a positive real number ε < 1, we say w-weighted Dirichlet s theorem is ε-improvable for x R 2 if (1) admits integer solutions (p, q) Z 2 Z for T sufficiently large. Denote by Sing(w) the set of w-singular vectors. Let DI(w, ε) be the set of vectors x R 2 for which w-weighted Dirichlet s theorem is ε-improvable. Then Sing(w) = DI(w, ε). 0<ε<1 Questions : What are the sizes of DI(w, ε) and Sing(w)? Lingmin LIAO, Université Paris-Est Hausdorff dimension of weighted singular vectors in R 2 12/28

13 III. Dynamical interpretation Dani 1985 observed that w-singular vectors correspond to certain divergent trajectories in the space L 3 of unimodular lattices in R 3 with respect to the one-parameter semi-group A + = {a t = diag(e w1t, e w2t, e t ) : t 0}. More precisely, x R 2 is w-singular if and only if A + h(x)z 3 is divergent where h(x) = x 1 x Lingmin LIAO, Université Paris-Est Hausdorff dimension of weighted singular vectors in R 2 13/28

14 IV. Un-weighted singular vectors in R 2 Y. Cheung considered the un-weighted singular vectors, i.e., the case w 1 = w 2 = 1/2. Cheung (Ann. Math 2011) : The set Sing((1/2, 1/2)) { } (x 1, x 2) : ε > 0, Q 1, n Q, max{ nx 1, nx 2 } < ε has Hausdorff dimension 4/3. There exists C > 0 such that for all ε > 0 small enough one has dim H DI(w, ε) C ε. Cheung Chevallier (Duke J. Math 2016) : generalization to dimension d. Further, for all t > 1/2, for all ε > 0 small enough, dim H DI(w, ε) εt. Q 1 2 Lingmin LIAO, Université Paris-Est Hausdorff dimension of weighted singular vectors in R 2 14/28

15 Figure Lingmin LIAO, 1. The Université Hausdorff Paris-Est Hausdorff dimension of ofsingpwq. weighted singular vectors in R 2 15/28 V. Weighted singular vectors in R 2 2 Theorem (L Shi Solan Tamam (arxiv 2016)) Suppose w = (w 1, w 2 ) where w 1 w 2 > 0 and w 1 + w 2 = 1. Then the Hausdorff dimension of Sing(w) is w 1. dim H Singpwq 2 3{2 4{ {2 1 w 1

16 VI. ε-improvable w-weighted Dirichlet s theorem Theorem (L Shi Solan Tamam (arxiv 2016)) Let w = (w 1, w 2 ) where w 1 w 2 > 0 and w 1 + w 2 = 1. There exists C > 0 such that for all 0 < ε 2 5/w2 one has Remark : w 1 dim H DI(w, ε) w 1 + C ε. The constant C in Theorem is computable. In the un-weighted case, the upper bound is the same as Cheung while our method does not give good lower bound as Cheung Chevallier. Lingmin LIAO, Université Paris-Est Hausdorff dimension of weighted singular vectors in R 2 16/28

17 Lower bound Lingmin LIAO, Université Paris-Est Hausdorff dimension of weighted singular vectors in R 2 17/28

18 I. Fractal structure Tree : Fractal : Rooted tree : a connected graph T without cycles and with a distinguished vertex τ 0, called the root of T. We identify T with the set of vertices of the tree T. The set of vertices of height n is denoted by T n. τ T n is connected with a unique τ n 1 T n 1 : τ is a son of τ n 1. T (τ) : the set of sons of τ T. A rooted branch is a sequence of vertices {τ n } n N where τ n is a son of τ n 1. Y be a Polish space. A fractal structure on Y : (T, β), where T is a rooted tree and β is a map from T to the set of nonempty compact subsets of Y. A fractal structure gives a fractal F(T, β) := {y Y : y n=0β(τ n ) for some rooted branch {τ n } n N }. Lingmin LIAO, Université Paris-Est Hausdorff dimension of weighted singular vectors in R 2 18/28

19 II. Regular self-affine structure We say that (T, β) is a regular fractal structure if each vertex of T has a nonempty set of sons ; if τ is a son of τ then β(τ) β(τ ) ; for any rooted branch {τ n } n N of T the diameter of β(τ n ) goes to zero as n tends to infinity. A rectangle means a rectangle in R 2 with sides parallel to axes. In particular, a rectangle with size l 1 l 2 and center x R 2 refers to the set {y R 2 : y 1 x 1 l 1 /2, y 2 x 2 l 2 /2}. A self-affine structure on R 2 is a fractal structure (T, β) on R 2 such that for every τ T the set β(τ) is a rectangle with size W (τ) L(τ). A self-affine structure is said to be regular if the corresponding fractal structure is regular. Lingmin LIAO, Université Paris-Est Hausdorff dimension of weighted singular vectors in R 2 19/28

20 III. Tool for lower bound Let (T, β) be a regular self-affine structure on R 2. Suppose there are sequences of positive real numbers {W n }, {L n }, {ρ n }, {C n } indexed by N {0} with the following properties : 1 W (τ) = W n, L(τ) = L n and W n L n for all n and τ T n ; 2 C 0 = 1 and T (τ) C n for all n N and τ T n 1 ; 3 ρ n 1 for all n. Moreover, for all τ n T n and different τ, κ T (τ n ) dist (β (τ), β (κ)) ρ n+1 W n. Let P n = n C k, D n = max {k n : L k W n}, k=0 s = sup t > 0 : ( ) log P nwnρ t t lim n+1 D n i=n+1 ρici n max{d n n, 1} =. Lemma If s > 1, then dim H F(T, β) s. Lingmin LIAO, Université Paris-Est Hausdorff dimension of weighted singular vectors in R 2 20/28

21 IV. Fractal structure for the lower bound Let 0 < ε < r be small enough. Let t = 100/ε. Take ε n = ε n, and t n such that t n t n 1 = nt for n N. For x = (x 1, x 2 ) R 2 and r 1, r 2 > 0, let I(x; r 1, r 2 ) = [x 1 r 1, x 1 + r 1 ] [x 2 r 2, x 2 + r 2 ] R 2. L 3 : the space of unimodular lattices in R 3. L 3 := {Λ L 3 : Λ Re 3 = rze 3 for some r 0 with 1/2 < r 0 1}. Define T as follows : τ 0 = (0, 0) is the root of T and for each n, T (κ) (κ T n 1 ) consists of such that : τ β(κ) := I(κ; ε n e w1tn tn 1, ε n e w2tn tn 1 ) a tn h(τ)z 3 L 3, and a tn h(τ)z 3 K ε 2 n, b n a tn h(τ)z 3 K r, with b n := diag(e w2nt, e w2nt, 1), K ε := {Λ L 3 : ϕ ε, ϕ Λ \ {0}}. Define β(τ) = I(τ; ε n e w1tn+1 tn, ε n e w2tn+1 tn ). Lingmin LIAO, Université Paris-Est Hausdorff dimension of weighted singular vectors in R 2 21/28

22 V. Counting numbers and distances Lemma F(T, β) Sing(w). Lemma For every n N and y T n 1 one has Lemma ε2 ne 2nt T (y) 10ε 2 ne 2nt. Let τ T n 1 (n N). Then for all different x, y T (τ) one has dist (β(x), β(y)) W n 1 r 8ε n 1 min{e w1nt, e (w1 w2)tn (1+w2)nt }. Lingmin LIAO, Université Paris-Est Hausdorff dimension of weighted singular vectors in R 2 22/28

23 Upper bound Lingmin LIAO, Université Paris-Est Hausdorff dimension of weighted singular vectors in R 2 23/28

24 I. Fractal relation Q : a countable set. A subset σ of Q 2 = Q Q is a relation on Q. For each τ Q, let σ(τ) = {κ Q : (τ, κ) σ}. A pair (τ, κ) Q 2 is said to be related by σ if there exist τ 1,..., τ n Q such that τ 1 = τ, τ n = κ and (τ i, τ i+1 ) σ for all 1 i < n. A triple (Q, σ, β) is said to be a fractal relation on a Polish space Y if β is a map from Q to nonempty compact subsets of Y σ is a relation on Q such that τ κ for any related pair (τ, κ). The triple (Q, σ, β) is admissible if for any sequence {τ n } n N with (τ n, τ n+1 ) σ one has diam β(τ n ) 0 as n. A fractal relation (Q, σ, β) on Y gives countably many fractal structures {(T [τ0], β)} τ0 Q where T [τ0] is a rooted tree with root τ 0 and for each τ T [τ0], T [τ0] (τ) = σ(τ). A fractal relation (Q, σ, β) gives a fractal set : F(Q, σ, β) := F(T [τ0], β). τ 0 Q Lingmin LIAO, Université Paris-Est Hausdorff dimension of weighted singular vectors in R 2 24/28

25 II. Tool for upper bound Lemma Let (Q, σ, β) be an admissible fractal relation on R 2 such that for every τ Q the compact set β(τ) is a rectangle with size W (τ) L(τ) where W (τ) L(τ). Suppose s is a positive real number with L(κ) W (κ) s 1 L(τ) W (τ) s 1 (2) κ σ(τ) for all τ Q, then dim H F(Q, σ, β) s. Lingmin LIAO, Université Paris-Est Hausdorff dimension of weighted singular vectors in R 2 25/28

26 III. Best approximation vector w-weighted quasi-norm : (x 1, x 2 ) w = max{ x 1 1/w1, x 2 1/w2 }. for u = (p, q) Z 2 N, write û = p q and u = q. We say (p, q) Z 2 N is a w-best approximation vector of x R 2 if 1 qx p w < q x p w for any (p, q ) Z 2 N with q < q ; 2 qx p w qx p w for any p Z 2. For every x R 2 \ Q 2 we associate a sequence Σ x = {u i } i N Z 2 N of w-best approximates of x with the following properties : u 1 > 1 ; u i < u i+1 for all i N ; there is no w-best approximate (p, q) of x with u i < q < u i+1. Lingmin LIAO, Université Paris-Est Hausdorff dimension of weighted singular vectors in R 2 26/28

27 IV. Best approximation vector and singular vectors For x R 2 and u = (p, q), let A(x, u) = qx p w. r(u) := min v Z 2 N,v u A(û, v). Lemma Let x R 2 \ Q 2 and Σ x = {u i } i N. Then x Sing(w) if and only if r(u i ) u i 0. Q ε = {u Z 2 N : r(u) u < ε, u > 1} u Z 2 N satisfies u u /2 and r(u) = A(û, u ). H u : hyperplane in R 3 generated by u and u. D(u, ε) = {v H u Q ε : v u, A( v, u) < 2 2/w2 r(u)}. For every v D(u, ε) let E(u, v, ε) = {w Q ε : w > v, w H u and A(ŵ, v) < ε w 1 }. We define σ ε (u) = E(u, v, ε). v D(u,ε) β(u) := I(û; u (w1+1), u (w2+1) ) Lingmin LIAO, Université Paris-Est Hausdorff dimension of weighted singular vectors in R 2 27/28

28 V. Essential part is covered We need only to estimate the upper bound for Sing(w) := {x Sing(w) : 1, x 1, x 2 are linearly independent over Q} since all the x R 2 with 1, x 1, x 2 linearly dependent over Q are singular and the Hausdorff dimension of Sing(w) \ Sing(w) is one. Lemma For every 0 < ε < 1 one has Sing(w) F(Q ε, σ ε, β). Fact : Write t = (s 1)(w 1 + 1) + w We have w σ ε(u) Lemma L(w) W (w) s 1 L(u) W (u) s 1 w σ ε(u) ( ) t u 1. w Let 0 < ε 2 2/w2. For all u Q ε, v D(u, ε) and t > 2 one has w E(u,v,ε) ( ) t v ε w t 2. Lingmin LIAO, Université Paris-Est Hausdorff dimension of weighted singular vectors in R 2 28/28