Uniform Diophantine approximation related to b-ary and β-expansions

Transcription

1 Uniform Diophantine approximation related to b-ary and β-expansions Lingmin LIAO (joint work with Yann Bugeaud) Université Paris-Est Créteil (University Paris 12) Universidade Federal de Bahia April 8th 2014 Lingmin LIAO, Université Paris-Est Créteil (Paris 12) Uniform Diophantine approximation related to b-ary and β-expansions 1/31

2 Outline 1 The problem 2 β-transformation 3 Results 4 Ideas and proofs Lingmin LIAO, Université Paris-Est Créteil (Paris 12) Uniform Diophantine approximation related to b-ary and β-expansions 2/31

3 I. Dirichlet Denote by the distance to the nearest integer. Asymptotic Dirichlet Theorem (1842) : 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. Uniform Dirichlet Theorem (Stronger than the asymptotic version) : 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. Lingmin LIAO, Université Paris-Est Créteil (Paris 12) Uniform Diophantine approximation related to b-ary and β-expansions 3/31

4 II. Approximation with a higher speed For w > 1, what is the size of the set L w := { θ : nθ < n w for infinitely many n }? By Borel-Cantelli Lemma, if w > 1, it is of Lebesgue measure 0. 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 : Continued fraction theory. Lingmin LIAO, Université Paris-Est Créteil (Paris 12) Uniform Diophantine approximation related to b-ary and β-expansions 4/31

5 III. Asympototic approximation-lebsgue measure Khintchine 1924 : Let Ψ : N R + be such that n nψ(n) is decreasing. Set : L Ψ := {θ : nθ < Ψ(n) i.o. n} L Ψ is of Lebsgue measure zero if Ψ(n) < ; L Ψ is of full Lebsgue measure if Ψ(n) =. Duffin-Schaefer 1941 Conjecture : If Ψ is not decreasing, then L Ψ is of full Lebsgue measure 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) Uniform Diophantine approximation related to b-ary and β-expansions 5/31

6 IV. Asympototic approximation-hausdorff dimension For a Lebesgue measure zero set, we will use Hausdorff dimension (denoted by dim H ) to describe the size of the set. Jarník 1929, Besicovith 1934 : For w > 1, dim H (L w ) = dim H { θ : nθ < n w i.o. n } = 2/(w + 1). Levesley 1998 : Ψ : N R + a decreasing function. Y R l, n Z m non-zero, and n := max{ n i }. Define : L Ψ (Y ) := { Θ R ml : nθ Y < Ψ( n ) i.o. n Z m \ {0} }. log Ψ(r) Denote λ := lim inf r r. For all Y R l, (m 1)l + m + l if λ > m dim H(L Ψ(Y )) = λ + 1 l ml if λ m l Lingmin LIAO, Université Paris-Est Créteil (Paris 12) Uniform Diophantine approximation related to b-ary and β-expansions 6/31

7 V. Recall of Hausdorff dimension Suppose E R d, {U 1, U 2,...} is called a δ-covering if E U i and diam(u i ) δ. Let s 0, H s δ (E) := inf { i=1 U i s : {U i } i 1 δ covering} Hδ s (E) is increasing with respect to δ s-hausdorff measure : H s (E) := lim δ 0 H s δ(e) Hausdorff dimension (Felix Hausdorff 1918) : dim H (E) := inf{s 0 : H s (E) = 0} = sup{s 0 : H s (E) = } Lingmin LIAO, Université Paris-Est Créteil (Paris 12) Uniform Diophantine approximation related to b-ary and β-expansions 7/31

8 VI. Dynamical setting-1 Let T θ be the rotation on R/Z. The sets we were studying are and {θ : T n θ (0) 0 < n w, i.o. n}, {θ : N 1, T n θ (0) 0 < N w has a solution 1 n N}. In general, consider a family of transformations {T θ } θ Θ (Θ R) on a metric space (X, d) (rotations, beta transformations...). Let r n be a sequence decreasing to 0. Fix x, y X, we could study the sets : L(x, y) := {θ Θ : d(t n θ x, y) < r n i.o. n}, and U(x, y) := {θ Θ : N 1, d(t n θ x, y) < r N has a solution 1 n N}. For β-transformation T β : the asymptotic case : Persson-Schmeling 2008, Li-Persson-Wang-Wu 2014 ; the uniform case : the present talk. Lingmin LIAO, Université Paris-Est Créteil (Paris 12) Uniform Diophantine approximation related to b-ary and β-expansions 8/31

9 VII. Dynamical setting-2 T a system on a metric space (X, d), and r n 0. Shrinking targets problems : find the sizes of the sets : 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, Kim... For the dimension of S(y) : Hill-Velani 1995, 1999, Urbański 2002, Fernández-Melián-Pestana 2007, Shen-Wang Dynamical Diophantine Approximation : find the sizes of the sets : D(x) :={y X : d(t n x, y) < r n i.o.}. Fan-Schmeling-Troubetzkoy 2013 for the doubling map and L-Seuret 2012 for Markov piecewise interval maps. The present talk considers uniform version of shrinking targets problems for a β-transformation T β. L-Kim : uniform version of dyn. Dioph. Appr. for an irrational rotation. Lingmin LIAO, Université Paris-Est Créteil (Paris 12) Uniform Diophantine approximation related to b-ary and β-expansions 9/31

10 VIII. Relation with the hitting time Let (T θ ) θ Θ (Θ R) be a family of systems on a metric space (X, d). Define and R θ (x, y) := lim inf r 0 We have (fixing x, y X) and 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 {θ Θ : d(t n θ x, y) < n v i.o.} {θ : R θ (x, y) 1/v}, {θ : d(t n θ x, y) < N v has a solution 1 n N} {θ : R θ (x, y) 1/v}. The same thing holds when fixing (θ, x) or (θ, y). Lingmin LIAO, Université Paris-Est Créteil (Paris 12) Uniform Diophantine approximation related to b-ary and β-expansions 10/31

11 β-transformation Lingmin LIAO, Université Paris-Est Créteil (Paris 12) Uniform Diophantine approximation related to b-ary and β-expansions 11/31

12 I. The β-transformation Rényi s β-transformation T β : [0, 1) [0, 1) is defined by Let Then d β,1 (x) = βx, x = βx β T β (x) := βx βx. d β,n (x) = d β,1 (T n 1 β (x)) for n 1. + T βx β = d β,1 β + d β,2 β 2 + d β,3 β 3 +. Sequence d β (x) = d β,1 (x)d β,2 (x) β-expansion of x. Expansion of 1 : can be obtained by extending : T β (1) = β β. Example : β = , expansion of 1 = 110 = = 1 β + 1 β β β 4 +. Lingmin LIAO, Université Paris-Est Créteil (Paris 12) Uniform Diophantine approximation related to b-ary and β-expansions 12/31

13 II. Dynamical properties Gel fond 1959, Parry 1960 :There is a unique (with a constant difference) measure h β (x)dx which is invariant under T β, where h β (x) = n 0, T n β (1)>x 1 β n. It is equivalent to the Lebesgue measure : (β 1)/β h β (x) β/(β 1). The entropy of a β-expansion is log β. Rohklin 1960 : T β is exact for all β > 1. Smorodinsky 1973 : T β is week Bernoulli for all β > 1. Hofbauer 1978 : T β admits a unique maximal measure for all β > 1. Lingmin LIAO, Université Paris-Est Créteil (Paris 12) Uniform Diophantine approximation related to b-ary and β-expansions 13/31

14 III. Parry s admissible sequence and β-shift A sequence a 1 a 2 is said admissible if x (0, 1], d β (x) = a 1 a 2. Lexicographical order : a 1 a 2 b 1 b 2 if and only if k 1, a i = b i for i < k and a k < b k. Denote a 1 a 2 b 1 b 2, if a 1 a 2 b 1 b 2 or a 1 a 2 = b 1 b 2. The β-shift S β on the alphabet {0, 1,..., β } is the closure of the set of admissible sequences. Theorem (Parry 1960) A sequence a 1 a 2 is admissible if and only if for each n 1 a n a n+1 d β(1). A sequence a 1 a 2 is in S β if and only if for each n 1 a n a n+1 d β(1). Here d β (1) := lim x 1 d β (x) called the infinite β-expansion of 1. Lingmin LIAO, Université Paris-Est Créteil (Paris 12) Uniform Diophantine approximation related to b-ary and β-expansions 14/31

15 IV. Dynamics of β-shift The infinite expansion d β (1) of 1 determines the whole shift. Bertrand 1986 : The sequence d β (1) is finite if and only if the beta-shift S β is a subshift of finite type. The sequence d β (1) is eventually periodic if and only if the beta-shift S β is sofic. The length of strings of zero s in d β (1) is bounded iff S β satisfies specification property : there exists k N such that for all admissible v, w, there exists u with length less than k and vuw is also admissible. The sequence d β (1) does not contains some admissible words iff S β is synchronizing : containing a synchronizing word u, i.e., for all v, w, if vu, wu are admissible, then vus is admissible wus is admissible Lingmin LIAO, Université Paris-Est Créteil (Paris 12) Uniform Diophantine approximation related to b-ary and β-expansions 15/31

16 Results Lingmin LIAO, Université Paris-Est Créteil (Paris 12) Uniform Diophantine approximation related to b-ary and β-expansions 16/31

17 I. Two exponents in b-ary expansion ξ be an irrational real number, b be an integer with b 2. Let v b (ξ) be the supremum of the real numbers v such that b n ξ < (b n ) v, i.o. n. Let ˆv b (ξ) be the supremum of the real numbers ˆv such that N 1, b n ξ < (b N ) ˆv has a solution 1 n N. Remark 1 : Consider T b : [0, 1] [0, 1] defined by T b ξ = bξ mod 1. Then 1/v b (ξ) = R(ξ, 0), 1/ˆv b (ξ) = R(ξ, 0) ( v b (ξ) ˆv b (ξ)). Remark 2 : The exponent v b (ξ) takes values in [0, ] while ˆv b (ξ) are in [0, 1]. Furthermore, v b (ξ) ˆv b(ξ) 1 ˆv b (ξ) (ˆv b (ξ) 1). Lingmin LIAO, Université Paris-Est Créteil (Paris 12) Uniform Diophantine approximation related to b-ary and β-expansions 17/31

18 II. Diophantine approximation related to b-ary expansion We study the sets : { ξ : b n ξ < (b n ) v for infinitely many n } = {ξ : v b (ξ) v} { ξ : N 1, b n ξ < (b N ) ˆv has a solution 1 n N } = {ξ : ˆv b (ξ) ˆv}, and further the set {ξ : ˆv b (ξ) = ˆv} {ξ : v b (ξ) = θˆv}, with θ 1 1 ˆv. Known results : from a general result of Borosh and Frankel 1972, we have : dim H {ξ : v b (ξ) v} = v, The mass transference principle of Beresnevich and Velani 2006 gives : dim H {ξ : v b (ξ) = v} = v. Lingmin LIAO, Université Paris-Est Créteil (Paris 12) Uniform Diophantine approximation related to b-ary and β-expansions 18/31

19 III. Uniform approximation related to b-ary expansion Theorem 1 (Bugeaud-L) Let b 2 be an integer. Let θ and ˆv be positive real numbers with ˆv < 1 and θ 1/(1 ˆv), then Furthermore, dim({ξ : ˆv b (ξ) = ˆv} {ξ : v b (ξ) = θˆv}) = dim{ξ : ˆv b (ξ) = 1} = 0. θ 1 θˆv (1 + θˆv)(θ 1). (1) Maximizing (1) with respect to θ [1/(1 ˆv), + ), we have a maximal value (1 ˆv) 2 /(1 + ˆv) 2 at the point θ 0 := 2/(1 ˆv) which implies the following theorem. Theorem 2 (Bugeaud-L) Let b 2 be an integer and ˆv be a real number in [0, 1]. Then we have dim{ξ : ˆv b (ξ) ˆv} = dim{ξ : ˆv b (ξ) = ˆv} = ( ) 2 1 ˆv. 1 + ˆv Lingmin LIAO, Université Paris-Est Créteil (Paris 12) Uniform Diophantine approximation related to b-ary and β-expansions 19/31

20 III. Our results on β-expansion x [0, 1], β > 1 be a real number. Let v β (x) be the supremum of the real numbers v such that T n β (x) < (β n ) v, i.o. n Let ˆv β (x) be the supremum of the real numbers ˆv such that N 1, T n β (x) < (β N ) ˆv has a solution 1 n N. Theorem 3 (Bugeaud-L) Let β > 1 be a real number. Let θ and ˆv be positive real numbers with ˆv < 1 and θ 1/(1 ˆv), then dim({x : ˆv β (x) = ˆv} {x : v β (x) = θˆv}) = Furthermore, dim({x : ˆv β (x) = 1}) = 0. and dim{x : ˆv β (x) ˆv} = dim{x : ˆv β (x) = ˆv} = θ 1 θˆv (1 + θˆv)(θ 1). ( 1 ˆv ) ˆv Lingmin LIAO, Université Paris-Est Créteil (Paris 12) Uniform Diophantine approximation related to b-ary and β-expansions 20/31

21 IV. Our results on β-expansion - continued Theorem 4 (Bugeaud-L) Let θ and ˆv be positive real numbers with ˆv < 1 and θ 1/(1 ˆv), then dim({β > 1 : ˆv β (1) = ˆv} {β > 1 : v β (1) = θˆv}) = Furthermore, dim({β > 1 : ˆv β (1) = 1}) = 0, and Remark : Shen-Wang 2013 : Persson-Schmeling 2008 : dim{β > 1 : ˆv β (1) ˆv} = ( 1 ˆv ) ˆv dim{x [0, 1] : v β (x) v} = v. dim{β > 1 : v β (1) v} = v. θ 1 θˆv (1 + θˆv)(θ 1). Lingmin LIAO, Université Paris-Est Créteil (Paris 12) Uniform Diophantine approximation related to b-ary and β-expansions 21/31

22 Ideas and proofs Lingmin LIAO, Université Paris-Est Créteil (Paris 12) Uniform Diophantine approximation related to b-ary and β-expansions 22/31

23 I. Upper bound-1 Let ξ := ξ + a j b j j 1 (b-ary expansion). Define the increasing sequences (n k ) k 1 and (m k ) k 1 as follows : for k 1, we have either a n k > 0, a n k +1 =... = a m k 1 = 0, a m k > 0 or a n k < b 1, a n k +1 =... = a m k 1 = b 1, a m k < b 1. Take the maximal subsequences (n k ) k 1 and (m k ) k 1 of (n k ) k 1 and (m k ) k 1, respectively, in such a way that the sequence (m k n k ) k 1 is non-decreasing. We have v b (ξ) > 0 m k n k (k ). Lingmin LIAO, Université Paris-Est Créteil (Paris 12) Uniform Diophantine approximation related to b-ary and β-expansions 23/31

24 II. Upper bound-2 Note that b n k m k < b n k ξ < b n k m k +1. By construction, we have and v b (ξ) = lim sup k + m k n k n k = lim sup k + m k n k 1 ˆv b (ξ) lim inf k + By the fact m k n k n k+1 ( lim sup k + ˆv b (ξ) 1 lim inf k + ) n k m k Thus (providing that ˆv b (ξ) < 1) m k n k m k = 1 lim sup k + n k m k. ( lim sup k + m k n k ) 1, we deduce v b (ξ) = v b(ξ) 1 + v b (ξ). v b (ξ) ˆv b(ξ) 1 ˆv b (ξ). (2) Lingmin LIAO, Université Paris-Est Créteil (Paris 12) Uniform Diophantine approximation related to b-ary and β-expansions 24/31

25 III. Upper bound-3 By definition, m k n k (v b (ξ) + ε)n k, m k n k n k+1 (ˆv ε). Hence, (v b (ξ) + ε)n k (ˆv ε)n k+1. Consequently, there exist an integer n and a positive real number ε such that the sum of all the lengths of the blocks of 0 or b 1 in the prefix of length n k of the infinite sequence a 1 a 2... is, for k large enough, at least equal to ( ) ) 2 (ˆv ε)n k (1 + ˆv ε v b (ξ)+ɛ + ˆv ɛ v b (ξ)+ɛ +... n = n k (ˆv ε)(v b (ξ)+ɛ) v b (ξ) ˆv+2ɛ with v b (ξ) = θˆv. n n k ( ˆvvb (ξ) v b (ξ) ˆv ɛ ) = n k ( ) θˆv θ 1 ɛ, Lingmin LIAO, Université Paris-Est Créteil (Paris 12) Uniform Diophantine approximation related to b-ary and β-expansions 25/31

26 IV. Upper bound-4 There exists c > 0 such that m k 1 n k 1 cn k. Consequently, C, s.t. there are at most C log n k blocks of 0 (or of b 1) in the prefix of length n k of the infinite sequence a 1 a Obviously, there are at most n k possible choices for their first index, we have in total at most 2 C log n k n C log n k k = (2n k ) C log n k possible choices. For each of these choices, at least n k (θˆv/(θ 1) ε ) digits are prescribed (and are equal to 0 or b 1). Consequently, at most ( θˆv n k n k θ 1 ε ) + 1 = n k (1 + ε ) θ 1 θˆv θ 1 digits in the prefix a 1 a 2... a nk, and thus in a 1 a 2... a mk, are free. Lingmin LIAO, Université Paris-Est Créteil (Paris 12) Uniform Diophantine approximation related to b-ary and β-expansions 26/31

27 V. Upper bound-5 The set of real numbers whose b-ary expansion starts with a 1 a 2... a mk defines an interval of length b m k. For ε > 0, there are infinitely many indices k such that b m k b (1+θˆv)(1 ε )n k. Then, a standard covering argument shows that we have to consider the series (2N) C log N b N(1+ε )(θ 1 θˆv)/(θ 1) b (1+θˆv)(1 ε )Ns. N 1 The critical exponent s 0 such that the above converges if s > s 0 and diverges if s < s 0 is given by It then follows that s 0 = 1 + ε 1 ε θ 1 θˆv (1 + θˆv)(θ 1). dim({ξ : ˆv b (ξ) ˆv} {ξ : v b (ξ) = θˆv}) θ 1 θˆv (1 + θˆv)(θ 1). Lingmin LIAO, Université Paris-Est Créteil (Paris 12) Uniform Diophantine approximation related to b-ary and β-expansions 27/31

28 VI. Lower bound-1 ˆv be in (0, 1) and θ be a real number with θ 1 1 ˆv. Choose two sequences (m k ) k 1 and (n k ) k 1 such that n k < m k < n k+1 for k 1, and such that (m k n k ) k 1 is non-decreasing. Furthermore, we assume that and lim k + lim k + m k n k n k+1 m k n k n k = ˆv, = θˆv. An easy way to construct such sequences is to start with n k = θ k, m k = (θˆv + 1)n k, and then to make a small adjustment to guarantee that (m k n k ) k 1 is non-decreasing. Lingmin LIAO, Université Paris-Est Créteil (Paris 12) Uniform Diophantine approximation related to b-ary and β-expansions 28/31

29 VII. Lower bound-2 We consider the set of real numbers ξ in (0, 1) whose b-ary expansion ξ = a j j 1 b satisfies, for k 1, j and a nk = 1, a nk +1 =... = a mk 1 = 0, a mk = 1, a mk +(m k n k ) = a mk +2(m k n k ) = = a mk +t k (m k n k ) = 1, where t k is the largest integer such that m k + t k (m k n k ) < n k+1. Observe that, since t k < n k+1 m k m k n k 2ˆv, for k large enough, the sequence (t k ) k 1 is bounded. The collection of the above ξ is our Cantor type set E θ,ˆv. We verify that for any ξ E θ,ˆv, ˆv b (ξ) = ˆv and v b (ξ) = θˆv. Lingmin LIAO, Université Paris-Est Créteil (Paris 12) Uniform Diophantine approximation related to b-ary and β-expansions 29/31

30 VIII. Lower bound-3 Let n be a large positive integer. Denote by I n (a 1,..., a n ) the interval composed of the real numbers in (0, 1) whose b-ary expansion starts with a 1... a n. Define a Bernoulli measure µ on E θ,ˆv as follows. We distribute the mass uniformly. If there exists k 2 such that n k n m k, then define µ(i n (a 1,..., a n )) = b (n k 1 k 1 j=1 (mj nj+1+tj)). If there exists k 2 such that m k < n < n k+1, then define µ(i n (a 1,..., a n )) = b n+ k 1 j=1 (mj nj+1+tj)+m k n k +1+t, where t is the largest integer such that m k + t(m k n k ) n. Lingmin LIAO, Université Paris-Est Créteil (Paris 12) Uniform Diophantine approximation related to b-ary and β-expansions 30/31

31 IX. Lower bound-4 We have lim inf k log µ(i mk ) log I mk = lim inf k n k 1 k 1 j=1 (m j n j t j ) m k. Recalling that (t k ) k 1 is bounded and that (m k ) k 1 grows exponentially fast in terms of k, we have k 1 log µ(i mk ) j=1 lim inf = lim inf (n j+1 m j ). k log I mk k m k By definition, we see that m k m k+1 n k+1 lim = θˆv + 1, lim = θ, and lim = θ k n k k m k k m k θˆv + 1. Thus, by the Stolz-Cesàro Theorem, k 1 j=1 lim (n j+1 m j ) n k+1 m k = lim = θ 1 θˆv k m k k m k+1 m k (θ 1)(θˆv + 1). Hence, lim inf k log µ(i mk ) log I mk = θ 1 θˆv (θ 1)(θˆv + 1). Lingmin LIAO, Université Paris-Est Créteil (Paris 12) Uniform Diophantine approximation related to b-ary and β-expansions 31/31