Inhomogeneous Diophantine approximation with general error functions - PowerPoint PPT Presentation

Inhomogeneous Diophantine approximation with general error functions Lingmin LIAO & Micha� l RAMS Universit´ e Paris-Est Cr´ eteil Advances on fractals and related topics Chinese University of Hong Kong December 10th 2012 Lingmin LIAO, Universit´ e Paris-Est Cr´ eteil Inhomogeneous Diophantine approximation with general error functions 1/9

I. Classic results α an irrational real number. � · � the distance to the nearest integer. Minkowski (1907) : If y �∈ Z + α Z , then for infinitely many n ∈ Z , 1 � nα − y � < 4 | n | . Khintchine (1926) : For any real number y , there exist infinitely many n ∈ N satisfying the Diophantine inequalities : 1 √ � nα − y � < 5 n. Cassels (1950) : The following set E ( α, c ) is of full measure for any constant c > 0 : y ∈ R : � nα − y � < c � � E ( α, c ) := n for infinitely many n . Lingmin LIAO, Universit´ e Paris-Est Cr´ eteil Inhomogeneous Diophantine approximation with general error functions 2/9

II. About Hausdorff dimension Define n →∞ n θ � nα � = 0 } . ω ( α ) := sup { θ ≥ 1 : lim inf Remark that α is a Liouville number if ω ( α ) = ∞ . Bernik-Dodson 1999 : the Hausdorff dimension of the set � y ∈ R : � nα − y � < 1 � E γ ( α ) := for infinitely many n ( γ ≥ 1) , n γ satisfies ω ( α ) · γ ≤ dim H E γ ( α ) ≤ 1 1 γ . Bugeaud/Schmeling-Troubetzkoy 2003 : for any irrational α , dim H E γ ( α ) = 1 γ . Lingmin LIAO, Universit´ e Paris-Est Cr´ eteil Inhomogeneous Diophantine approximation with general error functions 3/9

III. With a general error function Let ϕ : N → R + be a function decreasing to zero. Consider the set E ϕ ( α ) := { y ∈ R : � nα − y � < ϕ ( n ) for infinitely many n } . For an increasing function ψ : N → R + , define the lower and upper orders at infinity by log ψ ( n ) log ψ ( n ) λ ( ψ ) := lim inf and κ ( ψ ) := lim sup . log n log n n →∞ n →∞ Denote 1 1 u ϕ := l ϕ := κ (1 /ϕ ) . λ (1 /ϕ ) The results of Bugeaud and Schmeling, Troubetzkoy imply the inequality l ϕ ≤ dim H ( E ϕ ( α )) ≤ u ϕ . Lingmin LIAO, Universit´ e Paris-Est Cr´ eteil Inhomogeneous Diophantine approximation with general error functions 4/9

IV. Some known results A.-H. Fan, J. Wu 2006 : (1) If α is of bounded type dim H ( E ϕ ( α )) = u ϕ (2) There exists a Liouville number α and an error function ϕ such that dim H E ϕ ( α ) = l ϕ < u ϕ . J. Xu preprint : (1) For any α , log q n lim sup − log ϕ ( q n ) ≤ dim H ( E ϕ ( α )) ≤ u ϕ , n →∞ where q n denotes the denominator of the n -th convergent of the continued fraction of α . (2) For any irrational number α with ω ( α ) = 1 , dim H ( E ϕ ( α )) = u ϕ . Lingmin LIAO, Universit´ e Paris-Est Cr´ eteil Inhomogeneous Diophantine approximation with general error functions 5/9

V. Our results Theorem (L-Rams 2012) For any α with ω ( α ) = w ∈ [1 , ∞ ] , we have � � l ϕ , 1 + u ϕ �� min u ϕ , max ≤ dim H ( E ϕ ( α )) ≤ u ϕ . 1 + w Corollary : If w ≤ 1 /u ϕ , then dim H ( E ϕ ( α )) = u ϕ . Example : Take w = 2 , u = 1 / 2 and l = 1 / 3 . Construct α such that for all n , q 2 n ≤ q n +1 ≤ 2 q 2 n . Define n − 1 /l , q − 1 /l if q u/l k − 1 < n ≤ q u/l � � ϕ ( n ) = max . k k By Corollary, log q n lim − log ϕ ( q n ) = l < u = dim H ( E ϕ ( α )) . n →∞ Thus the lower bound of Xu is not optimal. Lingmin LIAO, Universit´ e Paris-Est Cr´ eteil Inhomogeneous Diophantine approximation with general error functions 6/9

VI. Our results-continued Theorem (L-Rams) For any irrational α and for any 0 ≤ l < u ≤ 1 , with u > 1 /w , there exists a decreasing function ϕ : N → R + , with l ϕ = l and u ϕ = u , such that � � l, 1 + u dim H ( E ϕ ( α )) = max < u. 1 + w Theorem (L-Rams) Suppose 0 ≤ l < u ≤ 1 . There exists a decreasing function ϕ : N → R + , with l ϕ = l and u ϕ = u , such that for any α which is not a Liouville number, dim H ( E ϕ ( α )) = u. Lingmin LIAO, Universit´ e Paris-Est Cr´ eteil Inhomogeneous Diophantine approximation with general error functions 7/9

VII. Method → Upper bound is OK. We treat the lower bound. Let B ≥ 1 and suppose there exists { n i } such that log q n i +1 → B. log q n i Let { m i } be such that q n i < m i ≤ q n i +1 . By passing to subsequences, we suppose the limit log m i N := lim log q n i i →∞ exists. (Obviously, 1 ≤ N ≤ B .) Let K > 1 . Denote y ∈ R : || nα − y || < 1 � � 2 q − K E i := for some n ∈ ( m i − 1 , m i ] . n i Let ∞ ∞ ∞ � � � E := E i and F := E i . i =1 j =1 i = j Lingmin LIAO, Universit´ e Paris-Est Cr´ eteil Inhomogeneous Diophantine approximation with general error functions 8/9

VIII. Method-continued Proposition (L-Rams) If { n i } is increasing sufficiently fast then � 1 � N 1 �� dim H E = dim H F = min K , max K , . 1 + B − N Choose a sequence m i of natural numbers such that log m i lim − log ϕ ( m i ) = u ϕ , i →∞ and choose n i such that q n i < m i ≤ q n i +1 . Take K = N/u ϕ . We have E ⊂ E ϕ and then � � u ϕ 1 �� dim H E ϕ ≥ min u ϕ , max N , . 1 + B − N Optimize the above value for B ∈ [1 , w ] , N ∈ [1 , B ] . Lingmin LIAO, Universit´ e Paris-Est Cr´ eteil Inhomogeneous Diophantine approximation with general error functions 9/9