Diophantine approximation of fractional parts of powers of real numbers

Transcription

1 Diophantine approximation of fractional parts of powers of real numbers Lingmin LIAO (joint work with Yann Bugeaud, and Micha l Rams) Université Paris-Est Numeration 2015, Nancy May 20th 2015 Lingmin LIAO, Université Paris-Est Diophantine approximation of x n } 1/20

2 Outline 1 Distribution of fractional parts of powers of real numbers 2 Asymptotic approximation of x n } 3 Uniform approximation of x n } 4 Proofs on the uniform approximation of x n } Lingmin LIAO, Université Paris-Est Diophantine approximation of x n } 2/20

3 I. Equidistribution A sequence (u n ) in [0, 1] is equidistributed if for all interval [a, b] [0, 1], Card1 n N : u n [a, b]} lim = b a. N N Theorem (Weyl, 1916) : A sequence (u n ) is equidistributed if and only if for every complex-valued, 1-periodic continuous function f, lim N 1 N N f(u n ) = n=1 and, if and only if for all integer h 0, lim N 1 N 1 0 f(x)dx, N e 2iπhun = 0. n=1 Lingmin LIAO, Université Paris-Est Diophantine approximation of x n } 3/20

4 II. (Equi)-Distribution of x n } Denote } the fractional part of a real number. Weyl 1916 : Let x > 1 be a real number. Then for almost all real ξ, the sequence ξx n } is equidistributed. Koksma 1935 : Let ξ 0 be a real number. Then for almost all real x > 1, the sequence ξx n } is equidistributed. Denote by the distance to the nearest integer. Thue 1910 (Hardy 1919) : Let ξ 0 and x > 1 be two real numbers. If there exist real numbers C > 0 and 0 < ρ < 1 such that ξx n < Cρ n for all n 1, then x is an algebraic number. Pisot 1937 : Let ξ 0 and x > 1 be two real numbers such that ξx n 2 <. n=0 Then ξ Q(x) and x is a Pisot-Vijayaraghavan number : an algebraic integer > 1, whose Galois conjugates have module < 1. Lingmin LIAO, Université Paris-Est Diophantine approximation of x n } 4/20

5 Remark : Vijayaraghavan Lingmin LIAO, Université 1948 Paris-Est proveddiophantine that for approximation all δ of > x0, n } there 5/20 are III. Sizes of exceptional sets Pollington 1979 : Let x > 1 be a real number. The set of numbers ξ such that ξx n } is not dense (so not equidistributed), has Hausdorff dimension 1. Pollington 1980 : Let ξ 0 be a real number. For all δ > 0, the set } x > 1 : ξx n } [0, δ] for all n 1 has Hausdorff dimension 1. Thus, the set of numbers x > 1 such that ξx n } is not dense (so not equidistributed), has Hausdorff dimension 1. Bugeaud Moshchevitin 2012, Kahane 2014 : Let (b n ) be an arbitrary sequence in [0, 1], and δ > 0. The set } x > 1 : x n b n δ for all large n has Hausdorff dimension 1. Kahane s question : for X > 1 2δ, } dim H 1 < x < X : x n b n δ for all large n =? Candidate : log(2δx)/ log X.

6 IV. 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) := infs 0 : H s (E) = 0} = sups 0 : H s (E) = } Lingmin LIAO, Université Paris-Est Diophantine approximation of x n } 6/20

7 Asymptotic approximation of x n } Lingmin LIAO, Université Paris-Est Diophantine approximation of x n } 7/20

8 I. Dirichlet s theorems 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. Lingmin LIAO, Université Paris-Est Diophantine approximation of x n } 8/20

9 II. Measure result - asymptotic approximation of x n } Koksma 1945 : Let (ɛ n ) be a real sequence with 0 ɛ n 1/2 for all n 1. If ɛ n <, then for almost all x > 1, x n ɛ n only for finitely many n. If (ɛ n ) is non-increasing and ɛ n =, then for almost allx > 1 x n ɛ n for infinitely many n. Mahler Szekeres 1967 : for almost x > 1, lim n xn 1/n = 1. Lingmin LIAO, Université Paris-Est Diophantine approximation of x n } 9/20

10 III. Dimension result -asymptotic approximation of x n } For x > 1, put P (x) := lim inf n x n 1/n. Mahler Szekeres 1967 : P (x) = 0 x is transcendental. Remark that for b > 1, x > 1 : P (x) < 1/b } = x > 1 : x n < b n for infinitely many n }. Question : What is the size of x > 1 : P (x) < 1/b }? Bugeaud Dubickas 2008 : For all real number X > 1, and b > 1, dim H 1 < x < X : P (x) < 1/b } = Moreover, dim H x > 1 : P (x) < 1/b } = 1. log X log(bx). Proof : Mass transference principle (Beresnevich Velani 2006) : Leb(lim sup B(x n, r n )) = 1 H s (lim sup B(x n, r 1/s n )) =. Lingmin LIAO, Université Paris-Est Diophantine approximation of x n } 10/20

11 IV. Dimension result - asymptotic case (continued) Bugeaud L Rams, in preparation : a constructive proof for the result of Bugeaud Dubickas. In general, for an arbitrary real sequence (y n ), dim H 1 < x < X : x n y n < b n for infinitely many n } = log X log(bx). Thus for a sequence (z k ), dim H k=1 1 < x < X : x n z k < b n for infinitely many n } = log X log(bx). Hence, we also have dim H k=1 x > 1 : x n z k < b n for infinitely many n } = 1. Lingmin LIAO, Université Paris-Est Diophantine approximation of x n } 11/20

12 Uniform approximation of x n } Lingmin LIAO, Université Paris-Est Diophantine approximation of x n } 12/20

13 I. Uniform approximation question For y R, we are interested in the following uniform approximation E(y, τ) = x > 1 : N 1, x n y τ N has a solution 1 n N }. Furthermore, we also study E(y, τ, b) := x > 1 : z n y < b n for infinitely many n and N 1, z n y < τ N has a solution 1 n N}. Question : Hausdorff dimensions of E(y, τ) and E(y, τ, b)? Lingmin LIAO, Université Paris-Est Diophantine approximation of x n } 13/20

14 II. Results on uniform approximation of x n } Theorem (Bugeaud L Rams, in preparation) Suppose b = τ θ with θ > 1. Then for all y R, ( ) log X θ dim H E(y, τ, b) ]1, X[ θ 1 log τ log X + θ log τ. Maximizing with respect to θ (θ = 2 log X/(log X log τ)), we have ( ) ( ) 2 log X log τ dim H E(y, τ) ]1, X[. log X + log τ Corollary (Bugeaud L Rams, in preparation) For all y R, and all b τ > 1, dim H E(y, τ, b) = dim H E(y, τ) = 1. Lingmin LIAO, Université Paris-Est Diophantine approximation of x n } 14/20

15 III. Some known results on uniform approximation-1 For an irrational θ, define w(θ) := supβ > 0 : lim inf j j β jθ = 0}. Let q n } be the denominators of continued fractions convergents of θ and U w [θ] := y : Q 1, 1 n Q, nθ y < Q w }. Theorem (D.H. Kim L, in preparation) Let θ be an irrational with w(θ) 1. Then the Hausdorff dimension of U w [θ] is 0 if w > w(θ), is 1 if w < 1/w(θ), and equals to lim k lim k log( k 1 j=1 (n1/w j n j θ ) n 1/w+1 log(n k n k θ 1 ) log( k 1 j=1 n j n j θ 1/w ) log(n k n k θ 1 ) k ), 1 w(θ) < w < 1,, 1 < w < w(θ). where n k is the (maximal) subsequence of (q k ) such that n 1/w k n k θ < 1, if 1/w(θ) < w < 1, n k n k θ 1/w < 2, if 1 < w < w(θ). Lingmin LIAO, Université Paris-Est Diophantine approximation of x n } 15/20

16 IV. Some known results on uniform approximation-2 Let β > 1 be a real number and T β be the β-transformation. Let v β (x) (resp. ˆv β (x)) be the supremum of the real numbers v (resp. ˆv) such that T n β (x) < (β n ) v, i.o. n, and respectively, N 1, T n β (x) < (β N ) ˆv has a solution 1 n N. Theorem (Bugeaud L, to appear) 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}) = θ 1 θˆv (1 + θˆv)(θ 1), dim(β > 1 : ˆv β (1) = ˆv} β > 1 : v β (1) = θˆv}) = θ 1 θˆv (1 + θˆv)(θ 1), ( 1 ˆv ) 2, dimx : ˆv β (x) ˆv} = dimx : ˆv β (x) = ˆv} = 1 + ˆv ( 1 ˆv ) 2. dimβ > 1 : ˆv β (1) ˆv} = 1 + ˆv Lingmin LIAO, Université Paris-Est Diophantine approximation of x n } 16/20

17 Proofs on the uniform approximation of x n } Lingmin LIAO, Université Paris-Est Diophantine approximation of x n } 17/20

18 I. Construction of a subset We will construct a subset F of the set E(X, y, τ, b) := 1 < x < X : z n y < b n for infinitely many n and N 1, z n y < τ N has a solution 1 n N}. Suppose b = τ θ with θ > 1. Take n k = θ k. Consider the points 1 < z < X such that [ ] (m z n k y < b n k 1 ) 1 ( m, z + y n k 1 ) 1, m + y + n k b n k b n. k Then z E(X, y, β, b) = E(X, y, β, τ θ ). Level 1 : I n1 (m, y, b) := [(m + y b n1 ) 1/n1, (m + y + b n1 ) 1/n1 ], where m ]1, X n1 [ is an integer. Level 2 : for an interval [c, d] at level 1, his son-intervals are I n2 (m, y, b) := [(m + y b n2 ) 1/n2, (m + y + b n2 ) 1/n2 ] with m [c n2, d n2 ]. Lingmin LIAO, Université Paris-Est Diophantine approximation of x n } 18/20

19 II. Computer the numbers and the lengths By construction, for the fundamental intervals containing z F, we have each interval [c k, d k ] at level k contains at least m k (z) = 2n k+1 c n k+1 1 k /n k b n k d n k 1 k son-intervals at level k + 1. ( z θ 1 b ) θ k the distance between intervals at level k + 1 contained in the interval [c k, d k ] at level k is at least ɛ k (z) = (1 2/b n k+1 )/n k+1 d n k+1 1 k ( z θ 1 b ) θ k. Lingmin LIAO, Université Paris-Est Diophantine approximation of x n } 19/20

20 III. Local dimension The local dimension of z F is bounded from below by log(m 1 (z) m k 1 (z)) log m k (z)ɛ k (z) (θ 1) log z log b = θk 1 (θ 1) log bz θ k (θ 1) log z log b ε (k) (θ 1) log bz ( (θ 1) log z log b ) k 1 j=1 θk log θ k log bz with ε (k) 0 when k. Using the relation b = τ θ, we have a lower bound of the dimension of E(X, y, τ, b) : (θ 1) log X log b (θ 1) log bx = log X θ θ 1 log τ log X + θ log τ. Lingmin LIAO, Université Paris-Est Diophantine approximation of x n } 20/20