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 Créteil Huazhong University of Science and Technology, Wuhan July 10th 2017 Lingmin LIAO, Université Paris-Est Créteil Diophantine approximation of {x n } 1/20

2 Outline 1 Asymptotic approximation of {x n } 2 Bad approximation of {x n } 3 Uniform approximation of {x n } 4 Proofs on the uniform approximation of {x n } Lingmin LIAO, Université Paris-Est Créteil Diophantine approximation of {x n } 2/20

3 Asymptotic approximation of {x n } Lingmin LIAO, Université Paris-Est Créteil Diophantine approximation of {x n } 3/20

4 I. (Equi)-Distribution of {x n } A sequence (u n ) in [0, 1] is equidistributed if for all interval [a, b] [0, 1], Card{1 n N : u n [a, b]} lim = b a. N N Denote {x} the fractional part of x, and x := inf{ x n : n Z}. Koksma 1935 : For almost all real x > 1, the sequence {x n } is equidistributed. Pollington 1980 : For all δ > 0, the set { } x > 1 : {x n } [0, δ] for all n 1 has Hausdorff dimension 1. Thus, the set of numbers x > 1 such that {x 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. Lingmin LIAO, Université Paris-Est Créteil Diophantine approximation of {x n } 4/20

5 II. 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 Créteil Diophantine approximation of {x n } 5/20

6 III. 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 all x > 1 x n ɛ n for infinitely many n. Bugeaud Dubickas 2008 : For all real number v > 1, and b > 1, dim H { 1 < x < v : b n for infinitely many n } = log v log(bv). Moreover, dim H { x > 1 : x n b n for infinitely many n } = 1. 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 Créteil Diophantine approximation of {x n } 6/20

7 IV. Asymptotic approximation our result Let b > 1, and y = (y n ) n 1 be an arbitrary sequence in [0, 1]. Define E(b, y) := {x > 1 : x n y n < b n for infinitely many n}. Theorem (Bugeaud L Rams, in preparation) For each v E(b, y), we have Hence dim H E(b, y) = 1. lim dim H([v ɛ, v + ɛ] E(b, y)) = log v ɛ 0 log(bv). Lingmin LIAO, Université Paris-Est Créteil Diophantine approximation of {x n } 7/20

8 V. Asymptotic approximation generalization Consider a family of strictly positive increasing C 1 functions f = (f n ) n 1 on an open interval I R such that f n (x), f n(x) > 1 for all x I. For τ > 1, define E(f, y, τ) := {x I : f n (x) y n < f n (x) τ for infinitely many n}. For v I, put u(v) := lim sup n log f n (v) log f n(v), log f n (v) l(v) := lim inf n log f n(v). Assume a regularity condition lim r 0 lim sup n sup x y <r which guarantees the continuity of u and l. log f n(x) log f n(y) = 1, (1) Lingmin LIAO, Université Paris-Est Créteil Diophantine approximation of {x n } 8/20

9 VI. Asymptotic approximation generalization For non-linear functions f n, i.e., when f n is not of the form f n (x) = a n x + b n, we also need the following condition : Theorem (Bugeaud L Rams, in preparation) Assume (1) and (2). If for all x I, log f M := sup n+1(v) n 1 log f n(v) <. (2) ɛ > 0, f n(x) ɛ <, (3) n=1 then for any v I τu(v) lim ɛ 0 dimh([v ɛ, v + ɛ] E(f, y, τ)) τl(v). If f n are linear then we do not need to assume (2), and lim dim 1 H([v ɛ, v + ɛ] E(f, y, τ)) = ɛ τl(v). Lingmin LIAO, Université Paris-Est Créteil Diophantine approximation of {x n } 9/20

10 Bad approximation of {x n } Lingmin LIAO, Université Paris-Est Créteil Diophantine approximation of {x n } 10/20

11 I. Bad approximation of {x n } Consider a family of C 1 strictly positive increasing functions f = (f n ) n 1 on an open interval I R such that f n (x), f n(x) > 1 for all x I and for all n 1. Let δ = (δ n ) n 1 be a sequence of positive real numbers such that δ n < 1/4. Define G(f, y, δ) := {x I : f n (x) y n δ n, n 1}. We need the following hypotheses : ɛ > 0, n 1, inf x (v ɛ,v+ɛ) f n+1(x) sup x (v ɛ,v+ɛ) f n(x) δn 2, (4) x I, log f lim n+1(x) =. (5) n log f n(x) Theorem (Bugeaud L Rams, in preparation) Under the hypotheses (1), (4) and (5), for all v G(f, y, δ), we have lim dim H([v ɛ, v + ɛ] G(f, y, δ)) = lim inf ɛ 0 n log f n(v) + n 1 j=1 log δ j log f n(v) log δ n. Lingmin LIAO, Université Paris-Est Créteil Diophantine approximation of {x n } 11/20

12 II. Bad approximation of {x n } continued Remark our result generalizes a recent result of S. Baker : f n (x) = x qn with (q n ) n 1 being a strictly increasing sequence of real numbers such that lim n (q n+1 q n ) =. Then dim H {x > 1 : lim n xqn y n = 0} = 1. Corollary (Bugeaud L Rams, in preparation) Consider a sequence of positive real numbers (a n ) n 1. If a n+1 lim =, n a n then for any sequence (y n ) n 1 of real numbers, we have dim H {x R : lim n a nx y n = 0} = 1. Lingmin LIAO, Université Paris-Est Créteil Diophantine approximation of {x n } 12/20

13 Uniform approximation of {x n } Lingmin LIAO, Université Paris-Est Créteil Diophantine approximation of {x n } 13/20

14 I. Uniform approximation of {nθ} For an irrational θ, define w(θ) := sup{β > 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, arxiv 2016) 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 Créteil Diophantine approximation of {x n } 14/20

15 II. Uniform approximation of beta-expansion 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, ETDS 2016) 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, dim{x : ˆv β (x) ˆv} = dim{x : ˆv β (x) = ˆv} = 1 + ˆv ( 1 ˆv ) 2. dim{β > 1 : ˆv β (1) ˆv} = 1 + ˆv Lingmin LIAO, Université Paris-Est Créteil Diophantine approximation of {x n } 15/20

16 III. Uniform approximation of {x n } For any sequence of real numbers y = (y n ) n 1 in [0, 1], define F (B, y) = { x > 1 : N 1, x n y n B N has a solution 1 n N }. Theorem (Bugeaud L Rams, in preparation) For any v F (B, y), we have lim dim H([v ɛ, v + ɛ] F (B, y)) ɛ 0 Hence dim H F (B, y) = 1. ( ) 2 log v log B. log v + log B Lingmin LIAO, Université Paris-Est Créteil Diophantine approximation of {x n } 16/20

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

18 I. Construction of a subset We will construct a subset F of the set F (v, ɛ, b, B, y) := {z [v ɛ, v + ɛ] : z n y n < b n for infinitely many n and N 1, z n y n < B N has a solution 1 n N}. Suppose b = B θ with θ > 1. Take n k = θ k. Consider the points z [v ɛ, v + ɛ] such that [ (m z n k y n < b n k m, z + yn 1 ) 1 ( n k, m + yn b n + 1 ] ) 1 n k k b n. k Then z F (v, ɛ, b, B, y) = F (v, ɛ, B θ, B, y) = F (v, ɛ, b, b 1 θ, y). Level 1 : I n1 (m, y n1, b) := [(m + y n1 b n1 ) 1/n1, (m + y n1 + b n1 ) 1/n1 ], where m ](v ɛ) n1, (v + ɛ) n1 [ is an integer. Level 2 : for an interval [c, d] at level 1, his son-intervals are I n2 (m, y n2, b) := [(m + y n2 b n2 ) 1/n2, (m + y n2 + b n2 ) 1/n2 ] with m [c n2, d n2 ]. Lingmin LIAO, Université Paris-Est Créteil 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 k-th level interval [c k (z), d k (z)] containing z, contains at least m k (z) = 2n k+1 c k (z) n k+1 1 /n k b n k d k (z) n k 1 son-intervals at level k + 1. the distance between intervals at level k + 1 contained in the interval [c k (z), d k (z)] is at least We have the estimations m k (z) ɛ k (z) = (1 2/b n k+1 )/n k+1 d k (z) n k+1 1. θ 2 θ+1 ( z θ 1 b ) θ k, m k (z)ε k (z) ( ) θ k θ+3 θ k. bz Lingmin LIAO, Université Paris-Est Créteil Diophantine approximation of {x n } 19/20

20 III. Local dimension The local dimension of z F is bounded from below by lim inf k = lim inf k log(m 1 (z) m k 1 (z)) log m k (z)ɛ k (z) (θ 1) log z log b (θ 1) log bz lim inf k θk 1 1 θ k 1 = ( (θ 1) log z log b ) k 1 j=1 θj θ k log bz (θ 1) log z log b. (θ 1) log bz Hence, by the relation b = B θ, we have the lower bound for the set F (v, ɛ, b, B, y) = F (v, ɛ, B θ, B, y) : (θ 1) log(v ɛ) log b (θ 1) log(b(v ɛ)) = log(v ɛ) θ θ 1 log B log(v ɛ) + θ log B. By maximizing the right side of the equality, with respect to θ > 1, we obtain the lower bound ( ) 2 log(v ɛ) log B. log(v ɛ) + log B Lingmin LIAO, Université Paris-Est Créteil Diophantine approximation of {x n } 20/20