Dirichlet uniformly well-approximated numbers

Transcription

1 Dirichlet uniformly well-approximated numbers Lingmin LIAO (joint work with Dong Han Kim) Université Paris-Est Créteil (University Paris 12) Hyperbolicity and Dimension CIRM, Luminy December 3rd 2013 Lingmin LIAO, Université Paris-Est Créteil (Paris 12) Dirichlet uniformly well-approximated numbers 1/33

2 Outline 1 The question 2 Metric Diophantine approximation 3 Dirichlet uniform approximation 4 Ideas and proofs Lingmin LIAO, Université Paris-Est Créteil (Paris 12) Dirichlet uniformly well-approximated numbers 2/33

3 I. Dirichlet Denote by the distance to the nearest integer. Uniform Dirichlet Theorem (1842) : 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. Asymptotic Dirichlet Theorem : 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. Lingmin LIAO, Université Paris-Est Créteil (Paris 12) Dirichlet uniformly well-approximated numbers 3/33

4 II. Our question Corresponding to Uniform Dirichlet Theorem, we study a question of Bugeaud and Laurent 2005 : for fixed θ, what is the size (Hausdorff dimension) of the following set U β [θ] := { y : Q 1, nθ y < Q 1/β } has a solution 1 n Q = lim inf Q n=1 ( B nθ, Q 1/β). Corresponding to Asympototic Dirichlet Theorem, there is a result of Bugeaud 2003, Troubetzkoy-Schmeling 2003 : For all θ R \ Q, β 1, define L β [θ] :={y : nθ y < n 1/β for infinitely many n} ( = lim sup B nθ, n 1/β). Then dim H (L β [θ]) = β. Lingmin LIAO, Université Paris-Est Créteil (Paris 12) Dirichlet uniformly well-approximated numbers 4/33

5 Metric Diophantine approximation Lingmin LIAO, Université Paris-Est Créteil (Paris 12) Dirichlet uniformly well-approximated numbers 5/33

6 I. General context (X, d) a complete metric space. {x n } n 1 a sequence of points in X. {r n } a sequence of real numbers. Define (i.o. = infinitely often) L({x n }, {r n }) := lim sup B(x n, r n ) = {y : d(y, x n ) < r n i.o.} Question : What is the size of L({x n }, {r n })? Its measure (if we have a measure µ on X)? Its Hausdorff dimension? Lingmin LIAO, Université Paris-Est Créteil (Paris 12) Dirichlet uniformly well-approximated numbers 6/33

7 II. Three examples X = R, {x n } = Q : { y : y p q < 1 q β } i.o.. X is a probability space and {x n } is a random sequence. Example : x n are i.i.d. { y : y x n < 1 } n β i.o.. (X, T ) is a dynamical system and x n = T n x. Example : X = R/Z, T x = 2x mod 1. { y : y T n x < 1 n β } i.o.. Lingmin LIAO, Université Paris-Est Créteil (Paris 12) Dirichlet uniformly well-approximated numbers 7/33

8 III. I.i.d. sequences X = T = R/Z. ω = {ω n } n 1 is an i.i.d. sequence uniformly distributed on T. r n a sequence of positive numbers decreasing to 0. We are interested in the following set : { L[ω] := y T : ω n y < r } n i.o. 2 ( = lim sup ω n r n n 2, ω n + r ) n 2 points covered by infinitely many random intervals ω n + ( rn 2, rn 2 ). Question (Dvoretzky 1956) : When almost surely L[ω] = T? Case r n = c/n : Kahane 1959 : L[ω] = T a.s. if c > 1. Erdös conjecture : L[ω] = T a.s. if and only if c 1. Billard 1965 : For c < 1 : No. Mandelbrot 1972, Orey (independently) : L[ω] = T a.s. if c = 1. Lingmin LIAO, Université Paris-Est Créteil (Paris 12) Dirichlet uniformly well-approximated numbers 8/33

9 IV. I.i.d. sequences (Continued) General case : Shepp 1972 : L[ω] = T a.s. if and only if n=1 1 n 2 er1+ +rn =. A dimension result : Fan-Wu 2004 : If r n = c/n κ,c > 0 and κ > 1, then almost surely dim H L[ω] = 1/κ. Other related works : Barral, Fan, Kahane, Jaffard, Jonasson-Steif... Lingmin LIAO, Université Paris-Est Créteil (Paris 12) Dirichlet uniformly well-approximated numbers 9/33

10 V. Dynamical sequences (X, T ) a dynamical system, x n = T n x. We are interested in the sets : L[x] :={y X : d(t n x, y) < r n i.o.}. Known results : Fan-Schmeling-Troubetzkoy 2013 for the doubling map and L-Seuret 2012 for Markov piecewise interval maps. Remarque : Different to the Shrinking targets problems. S(y) :={x X : d(t n x, y) < r n i.o.}. For the measure of S(y), there are Dynamical Borel-Cantelli Lemma studied by : Boshernitzan, Chernov-Kleinbock, Chazottes, Fayad, Galatalo-Rousseau-Saussol, Haydn-Nicol-Persson-Vaienti, Kim... For the dimension of S(y) : Hill-Velani 1995, 1999, Urbański 2002, Fernández-Melián-Pestana 2007, Shen-Wang 2011, Li-Wang-Wu-Xu Lingmin LIAO, Université Paris-Est Créteil (Paris 12) Dirichlet uniformly well-approximated numbers 10/33

11 VI. Asympototic Approximation-Lebsgue measure Khintchine : Let Ψ : N R + be a decreasing function. Consider : L Ψ := {θ : nθ < Ψ(n) i.o.} L Ψ is of Lebsgue measure zero if Ψ(n) < ; L Ψ is of Lebsgue measure full if Ψ(n) =. Duffin-Schaefer 1941 Conjecture : If Ψ is not decreasing, then L Ψ is of Lebsgue measure full if φ(n)ψ(n)/n =, where φ is the Euler function. Haynes-Pollington-Velani 2012 : Yes, if φ(n)( Ψ(n) n )1+ɛ =, with ɛ > 0 small. Beresnevich-Harman-Haynes-Velani 2013 : Yes, if φ(n) Ψ(n) n exp(c log log n)(log log log n) =, with c > 0. Lingmin LIAO, Université Paris-Est Créteil (Paris 12) Dirichlet uniformly well-approximated numbers 11/33

12 VII. Asympototic Approximation-Hausdorff Dimension Define L β (y) := {θ : nθ y < n 1/β i.o.} Jarník 1929, Besicovith 1934 : For β 1, dim H (L β (0)) = 2β/(β + 1). Levesley 1998 : For any y R, and β 1, Define dim H (L β (y)) = 2β/(β + 1). L β [θ] := {y : nθ y < n 1/β Bernik-Dodson 1999 : For all θ R \ Q, and β 1, ω β dim H (L β [θ]) β i.o.} where ω 1 is a real number such that nθ n 1/ω ev.. Bugeaud 2003, Troubetzkoy-Schmeling 2003 : For all θ R \ Q, β 1 : dim H (L β [θ]) = β. Lingmin LIAO, Université Paris-Est Créteil (Paris 12) Dirichlet uniformly well-approximated numbers 12/33

13 Dirichlet uniform approximation Lingmin LIAO, Université Paris-Est Créteil (Paris 12) Dirichlet uniformly well-approximated numbers 13/33

14 I. Uniformly approximated points In general, we consider the sizes of the following sets : { U β (0) := θ : Q 1, 1 n Q, nθ < Q 1/β}, U β (y) := and for fixed θ, Remark : U β [θ] := { θ : Q 1, 1 n Q, nθ y < Q 1/β}. { y : Q 1, 1 n Q, nθ y < Q 1/β}. U β (0) = { Q if β < 1 R if β 1. Lingmin LIAO, Université Paris-Est Créteil (Paris 12) Dirichlet uniformly well-approximated numbers 14/33

15 II. A known result in higher dimensional cases Theorem (Y. Cheung (Ann. Math 2011)) The set { (θ 1, θ 2 ) : δ > 0, Q 1, 1 n Q, max{ nθ 1, nθ 2 } < δ has Hausdorff dimension 4/3. Remark : The set is called singular points set. It has some important geometric meaning. Let G = SL 3 R, Γ = SL 3 Z. For t > 0, x = (x 1, x 2 ), let g t = et e t 0, h x = 1 0 x x e 2t The vector x is singular iff (g t h x Γ) t 0 is a divergent trajectory of homogeneous flow on G/Γ induced by g t. Q 1 2 }. Lingmin LIAO, Université Paris-Est Créteil (Paris 12) Dirichlet uniformly well-approximated numbers 15/33

16 III. Our result For an irrational θ, define w(θ) := sup{β > 0 : lim inf j j β jθ = 0}. Theorem (L, D.H. Kim) Let θ be an irrational with w(θ) 1. Then dim H (U β [θ]) equals to 1, β > w(θ), log(n β 1 lim n 1θ n β 2 n 2θ n β k 1 n k 1θ n β+1 k ) k log(n k n k θ 1, 1 < β < w(θ), ) log(n 1 n 1 θ β n 2 n 2 θ β n k 1 n k 1 θ β ) 1 lim k log(n k n k θ 1, ) w(θ) < β < 1, 0, β < 1 w(θ). where n k is the (maximal) subsequence of (q k ) such that { n β k n kθ < 1, if 1 < β < w(θ), n 1/β k n k θ < 1, if 1/w(θ) < β < 1. Lingmin LIAO, Université Paris-Est Créteil (Paris 12) Dirichlet uniformly well-approximated numbers 16/33

17 IV. Our result - Continued Theorem (L-D.H. Kim) For any irrational θ with w(θ) = w > 1 we have dim H (U β [θ]) = 1, β w, wβ 1 w 2 1 dimh (U β[θ]) β + 1 w + 1, 1 β < w, If w(θ) = 1, then we have 0 dim H (U β [θ]) wβ 1 w 2 1, 1 w < β < 1, dim H (U β [θ]) = 0, β 1 w. dim H (U β [θ]) = 1, β > 1, 1 2 dimh (U β[θ]) 1, β = 1, dim H (U β [θ]) = 0, β < 1. If w(θ) =, then dim H (U β [θ]) = 0. Lingmin LIAO, Université Paris-Est Créteil (Paris 12) Dirichlet uniformly well-approximated numbers 17/33

18 V. Graphs and comparing with the asymptotic case 1 2 w+1 1 w w 1 w β β case w(θ) > 1 case w(θ) = 1 Lingmin LIAO, Université Paris-Est Créteil (Paris 12) Dirichlet uniformly well-approximated numbers 18/33

19 VI. Remarks Remark 1 : The results depend on w(θ). Remark 2 : For the case β < 1, optimize the upper bound w.r.t. w(θ) : dim H (U β [θ]) β 2 2(1 + 1 β 2 ). Since β 2 > 1, we have dim H (U β [θ]) < β 2. Recall that for β < 1, U β [θ] L β [θ] except for a countable set of points. dim H (L β [θ]) = β. Remark 3 : No mass transference principe for uniform approximations. Remark 4 : Our results give an answer for the case of dimension one of a problem of Bugeaud and Laurent Lingmin LIAO, Université Paris-Est Créteil (Paris 12) Dirichlet uniformly well-approximated numbers 19/33

20 VII. Examples (i) Let θ be an irrational with w(θ) = w > 1 and q k+1 > qk w Then for each 1/w < β < w we have for all k. dim H (U β [θ]) = wβ 1 w 2 1. (ii) Assume that θ is an irrational with w(θ) = w > 1 and with the subsequence {k i } of q ki+1 > qk w i satisfying that a n+1 = 1 for n k i and q ki+1 > (q ki ) 2i. Then we have β + 1, for 1 β < w, dim H (U β [θ]) = w + 1 0, for β < 1. Lingmin LIAO, Université Paris-Est Créteil (Paris 12) Dirichlet uniformly well-approximated numbers 20/33

21 VIII. Examples-continued (iii) Let θ = 5 1 2, of which partial quotients a k = 1 for all k. Note that w(θ) = 1. Then U β [θ] = T for β = 1. Thus, we have { 1, β 1, dim H (U β [θ]) = 0 β < 1. (iv) Let θ be the irrational with partial quotient a k = k for all k. Then w(θ) = 1 and 1, β > 1, 1 dim H (U β [θ]) = 2, β = 1, 0 β < 1. Lingmin LIAO, Université Paris-Est Créteil (Paris 12) Dirichlet uniformly well-approximated numbers 21/33

22 Ideas and proofs Lingmin LIAO, Université Paris-Est Créteil (Paris 12) Dirichlet uniformly well-approximated numbers 22/33

23 I. Relation with the hitting time Let T θ be the rotation on R/Z. Define Define the hitting time rates : R θ (x, y) := lim inf r 0 We have τr θ (x, y) = inf{n : Tθ n x B(y, r)}. log τr θ (x, y), R θ (x, y) := lim sup log r r 0 log τr θ (x, y). log r L β (y) {θ : R θ (0, y) β}, U β (θ) {θ : R θ (0, y) β}, and L β [θ] {y : R θ (0, y) β}, U β [θ] {y : R θ (0, y) β}. Lingmin LIAO, Université Paris-Est Créteil (Paris 12) Dirichlet uniformly well-approximated numbers 23/33

24 II. A result on hitting time For an irrational θ, let w = w(θ) := sup{β > 0 : lim inf j jβ jθ = 0}. D. H. Kim and B. K. Seo 2003 : for every x and y lim sup r 0 log τ θ r (x, y) log r w, lim inf r 0 log τ r (x, y) log r 1. and for almost every x and y lim sup r 0 log τ θ r (x, y) log r = w, lim inf r 0 log τ θ r (x, y) log r = 1. Lingmin LIAO, Université Paris-Est Créteil (Paris 12) Dirichlet uniformly well-approximated numbers 24/33

25 III. Cantor structure Let Then we have G n = ( n ( ) ) 1/β 1 B iθ, n i=1 l=1 n=l and F k = U β [θ] = G n = F k. l=1 k=l q k+1 1 n=q k G n. We will show that for all l 1, the Hausdorff dimensions of k=l F k are the same. Take E n := n F k. k=1 Then E n is the union of the intervals at level n. Lingmin LIAO, Université Paris-Est Créteil (Paris 12) Dirichlet uniformly well-approximated numbers 25/33

26 IV. Calculation tools For each k, E k is a uion of finite intervals. Suppose E 0 E 1 E Let F = n=0 E n. Fact (Falconer s book p.64) Suppose each interval of E k 1 contains at least m k intervals of E k (k = 1, 2,... ) which are separated by gaps of at least ε k, where 0 < ε k+1 < ε k for each k. Then dim H (F ) lim k log(m 1 m k 1 ). log(m k ε k ) Fact (Falconer s book p.59) Suppose F can be covered by l k sets of diameter at most δ k with δ k 0 as k. Then log l k dim H (F ) lim. k log δ k Lingmin LIAO, Université Paris-Est Créteil (Paris 12) Dirichlet uniformly well-approximated numbers 26/33

27 V. Three Distances Theorem Given an irrational number θ and a positive integer N, if one arranges the points {θ}, {2θ},..., {Nθ} in ascending order, the distance between consecutive points can have at most three different lengths, and if there are three, one will be the sum of the other two. More precisely, Theorem (V. T. Sós 1958) If q k 1 + q k N < q k + q k+1, let N = q k 1 + q k + nq k + j, with 0 n a k+1 1 and 0 j q k 1. Then there are q k 1 + q k + nq k intervals of length q k θ, q k j intervals of length q k 1 θ n q k θ and k intervals of length q k 1 θ (n + 1) q k θ. Corollary If N = q k, for some k 0, then there are q k q k 1 intervals of length q k 1 θ and q k 1 intervals of length q k 1 θ + q k θ. Lingmin LIAO, Université Paris-Est Créteil (Paris 12) Dirichlet uniformly well-approximated numbers 27/33

28 VI. Structure of n-th level intervals-(i) Lemma If then we have F k = T. Facts : ( ) 1/β 1 2 q k 1θ + q k θ, (1) q k+1 If 1 < β < w(θ) and q β k q kθ 1, then (1) works, thus F k = T. If β = 1 and a k+1 = 1, then (1) works, thus F k = T. Lingmin LIAO, Université Paris-Est Créteil (Paris 12) Dirichlet uniformly well-approximated numbers 28/33

29 VII. Structure of n-th level intervals-(ii) Lemma For any β 1, we have F k = q k i=1 ( ) iθ q 1/β k+1, iθ + r k(i) where r k (i) equals to ( ( min a k+1 q k θ + q 1/β k+1, min (c 1)q k θ + (cq k + i 1) 1/β)), 1 c a k+1 i q k 1, ( ( min (a k+1 1)q k θ + q 1/β k+1, min (c 1)q k θ + (cq k + i 1) 1/β)), 1 c<a k+1 i > q k 1. Lingmin LIAO, Université Paris-Est Créteil (Paris 12) Dirichlet uniformly well-approximated numbers 29/33

30 VIII. Structure of n-th level intervals-(iii) Lemma For any β 1, we have q k i=1 ( ( iθ q 1/β k+1, iθ + C qk θ ) 1 ) 1+β β 2 q k θ q k F k where C β = β 1 1+β + β β 1+β. q k i=1 ( ( iθ q 1/β k+1, iθ + C qk θ ) 1 ) 1+β β, q k Lingmin LIAO, Université Paris-Est Créteil (Paris 12) Dirichlet uniformly well-approximated numbers 30/33

31 IX. Lower bound for 1 < β < w(θ) Suppose 1 < β < w(θ). Take (k i ) i 1 those that q β k i q ki < 1. Then Define F k = F ki. k=1 i=1 i E i = F kj, and F = E i. j=1 i=1 Let m i+1 be the number of intervals of E i+1 contained in E i. Then m i+1 q k i+1 2 ( qki θ q ki ) 1 1+β. Let ε i be the smallest gap between the intervals in E i. Then ε i 1 2 q k i 1θ. Using the formula in Falconer s book (page 64), we get the lower bound. Lingmin LIAO, Université Paris-Est Créteil (Paris 12) Dirichlet uniformly well-approximated numbers 31/33

32 X. Upper bound for 1 < β < w(θ) The set E i can be covered by l i sets of diameter at most δ i, with 1 l i q C β + 2 k1 1θ q k2 1θ ( qki θ δ i = (C β + 2) q ki ( qk1 θ ) 1 β+1 q k1 ) 1 β+1. C β + 2 ( qki 1 θ ) 1 β+1, q ki 1θ q ki 1 Using the formula in Falconer s book (page 59), we get the upper bound. Lingmin LIAO, Université Paris-Est Créteil (Paris 12) Dirichlet uniformly well-approximated numbers 32/33

33 XI. Some words for the other cases Lemma If β < 1, then for large q k, where c k = max(c k,1) q k i=1 ( B iθ, q 1/β k+1 ) F k (c k +2)q k i=1 ( B iθ, q 1/β k+1 ( qk θ β q k ) 1/(1+β). All intervals are disjoint. ). More efforts are needed for β = 1. For example, the following lemma. Lemma 1 If (b+1)(b+2) q k q k θ < 1 b(b+1), b 1, then we have ( iθ 1 ) 1, iθ + (b 1)q k θ + F k. q k+1 (b + 1)q k 1 i q k Lingmin LIAO, Université Paris-Est Créteil (Paris 12) Dirichlet uniformly well-approximated numbers 33/33