The dimensional theory of continued fractions Jun Wu Huazhong - PowerPoint PPT Presentation

Notation The dimensional theory of continued fractions Jun Wu Huazhong University of Science and Technology Advances on Fractals and Related Topics, Hong Kong, Dec. 10-14, 2012 Jun Wu, Huazhong University of Sci. & Tech. The dimensional theory of continued fractions

Notation Notation Continued fraction : x ∈ [0 , 1) , 1 x = 1 a 1 ( x ) + 1 a 2 ( x ) + a 3 ( x ) + · · · = [ a 1 ( x ) , a 2 ( x ) , a 3 ( x ) , · · · ] . Gauss Transformation : T : [0 , 1) → [0 , 1) given by � 1 T ( x ) := 1 � T (0) := 0 , x − for x ∈ (0 , 1) , x is called the Gauss transformation. Partial Quotients : For all n ∈ N , we have � 1 � � 1 � a 1 ( x ) = , a n ( x ) = . T n − 1 x x a n ( x ) ( n ≥ 1 ) are called the partial quotients of x . Jun Wu, Huazhong University of Sci. & Tech. The dimensional theory of continued fractions

Notation Convergents : Let p n ( x ) q n ( x ) = [ a 1 ( x ) , a 2 ( x ) , · · · , a n ( x )] , n ≥ 1 denote the convergents of x . n -order Basic Intervals : For any n ≥ 1 and ( a 1 , a 2 , · · · , a n ) ∈ N n , let I ( a 1 , a 2 , · · · , a n ) be the set of numbers in [0 , 1) which have a continued fraction expansion begins by a 1 , a 2 , · · · , a n . Gauss Measure : The Gauss measure G on [0 , 1) is given by 1 1 dG ( x ) = x + 1 dx. log 2 T preserves Gauss measure G and is ergodic with respect to G . Jun Wu, Huazhong University of Sci. & Tech. The dimensional theory of continued fractions

Notation Relation to Diophantine approximation Dirichlent’s Theorem, 1842 : Suppose x ∈ [0 , 1) \ Q . Then there exists infinitely many pairs p, q of relatively prime integers such that | x − p q | < 1 q 2 . Khintchine’s Theorem, 1924 : Let ϕ : R → R + be a continuous function such that x 2 ϕ ( x ) is not increasing. Then the set { x ∈ [0 , 1) : | x − p q | < ϕ ( q ) i . o . p q } ∞ � has Lebesgue measure zero if qϕ ( q ) converges and has full Lebesgue q =1 measure otherwise. Jun Wu, Huazhong University of Sci. & Tech. The dimensional theory of continued fractions

Notation Jarnik, 1929, 1931, Besicovitch, 1934 : For any β > 2 , x ∈ [0 , 1) : | x − p q | < 1 q β i . o . p = 2 � � dim H β . q Bugeaud, 2003 Bugeaud studied the sets of exact approximation order by rational numbers which significantly strengthens the result of Jarnik and Besicovitch. Lagrange’s Theorem : | x − p 1 ⇒ p q = p n ( x ) q | < 2 q 2 = q n ( x ) for some n. Moreover, ( a n +1 ( x ) + 2) q n ( x ) 2 ≤ | x − p n ( x ) 1 1 q n ( x ) | ≤ a n +1 ( x ) q n ( x ) 2 . Jun Wu, Huazhong University of Sci. & Tech. The dimensional theory of continued fractions

Notation Thus for any β > 2 , { x ∈ [0 , 1) : | x − p q | < 1 q β i . o . p q } { x ∈ [0 , 1) : | x − p n ( x ) 1 = q n ( x ) | < q n ( x ) β i . o . n } { x ∈ [0 , 1) : a n +1 ( x ) > q n ( x ) β − 2 i . o . n } ∼ We concluding : The growth speed of the partial quotients { a n ( x ) } n ≥ 1 reveals the speed how well a point can be approximated by rationals. In this talk, we shall concentrate on the sets of points whose partial quotients { a n ( x ) } n ≥ 1 have different growth speed. Jun Wu, Huazhong University of Sci. & Tech. The dimensional theory of continued fractions

Notation The growth speed of { a n ( x ) } n ≥ 1 Borel-Bernstein Theorem, 1912 : Let φ be an arbitrary positive function defined on natural numbers N and F ( φ ) = { x ∈ [0 , 1) : a n ( x ) ≥ φ ( n ) i.o. } . ∞ φ ( n ) converges, then L 1 ( F ( φ )) = 0 . If the series 1 � If the series n =1 ∞ 1 φ ( n ) diverges, then L 1 ( F ( φ )) = 1 . � n =1 Jun Wu, Huazhong University of Sci. & Tech. The dimensional theory of continued fractions

Notation Good, 1941 : dim H { x : a n ( x ) → ∞ , as n → ∞} = 1 2 . Hirst, 1970 : For any a > 1 , dim H { x ∈ [0 , 1) : a n ( x ) ≥ a n for any n ≥ 1 } = 1 2 . Luczak, 1997 : For any a > 1 , b > 1 , dim H { x : a n ( x ) ≥ a b n , ∀ n ∈ N } 1 dim H { x : a n ( x ) ≥ a b n , i . o . n ∈ N } = = b + 1 . Multifractal analysis : M. Pollicott, B. Weiss ; M. Kessebohmer, B. Stratmann ; A. Fan, L. Liao, Wang, Wu ; T. Jordan ; G. Iommi ; · · · Jun Wu, Huazhong University of Sci. & Tech. The dimensional theory of continued fractions

Notation Our intention For general ψ , what are dimensions of the sets � � E all ( ψ ) = x : a n ( x ) ≥ ψ ( n ) , ∀ n ∈ N and � � F i.o. ( ψ ) = x : a n ( x ) ≥ ψ ( n ) , i.o. n ∈ N ? Jun Wu, Huazhong University of Sci. & Tech. The dimensional theory of continued fractions

Notation Results : Wang and Wu (2008) 1 log log ψ ( n ) dim H E all ( ψ ) = 1 + b, b = lim sup , n n →∞ Wang and Wu (2008) Let log ψ ( n ) log log ψ ( n ) lim inf = log B, lim inf = log b. n n n →∞ n →∞ when 1 ≤ B < ∞ , dim H F i.o. ( ψ ) = inf { s ≥ 0 : P ( − s (log | T ′ | + log B )) ≤ 0 } , when B = ∞ , dim H F i.o. ( ψ ) = 1 / (1 + b ) , Jun Wu, Huazhong University of Sci. & Tech. The dimensional theory of continued fractions

Notation Good (1941) also considered the general set. For any B > 1 , let d B = dim H { x ∈ [0 , 1) : a n ( x ) ≥ B n i.o. } . He gave the bound estimation on its Hausdorff dimension only but not 2 , let θ ( s ) = 2 − s 3 − s + 3 − s 4 − s + · · · . Let the exact value. For any s > 1 s 0 satisfy θ ( s 0 ) = 1 . ∞ � n − s , Riemann’s zeta function. For any s > 1 , let ξ ( s ) = n =1 Good, 1941 : 2 < s < s 0 and B 4 s < θ ( s ) , then d B ≥ s . If 1 2 and B s ≥ ξ (2 s ) , then d B ≤ s . If s > 1 d B → 1 2 as B → ∞ . Jun Wu, Huazhong University of Sci. & Tech. The dimensional theory of continued fractions

Notation Sketch of Proof : { x : a n ( x ) ≥ B n , i.o.n } Upper bound : For any ( a 1 , · · · , a n ) , define basic interval : x : a 1 ( x ) = a 1 , · · · , a n ( x ) = a n , a n +1 ( x ) ≥ B n +1 � � J n ( a 1 , · · · , a n ) := . Then we get a cover of F i.o. = { x : a n ( x ) ≥ B n , i.o.n } : ∞ ∞ � � � F i.o. ⊂ J n ( a 1 , · · · , a n ) . N =1 n = N a 1 , ··· ,a n ∈ N So, the Hausdorff measure can be estimated as ∞ � s � 1 � � H s ( F i.o. ) ≤ lim inf . q 2 n ( a 1 , · · · , a n ) B n +1 N →∞ n = N a 1 , ··· ,a n ∈ N Jun Wu, Huazhong University of Sci. & Tech. The dimensional theory of continued fractions

Notation Lower bound : Define a Cantor subset : { n k } a largely sparse subsequence of N , α a large integer. define Cantor subset � � � 1 ≤ a n ( x ) ≤ α, when n � = n k ; E B ( α ) = x : a n ( x ) ∈ [ B n , 2 B n ) , when n = n k . Result : � s ≥ 0 : P α ( − s (log | T ′ | + log B )) = 0 � dim H E B ( α ) = inf , where 1 � exp( ψ ( x ) + · · · + ψ ( T n − 1 x )) . P α ( ψ ) = lim n log n →∞ 1 ≤ a 1 , ··· ,a n ≤ α Jun Wu, Huazhong University of Sci. & Tech. The dimensional theory of continued fractions

Notation Lemma (Mauldin & Urbanski, 1996) Let f : [0 , 1] → R be a function satisfying the tempered distortion condition, then P ( f ) = sup { P Σ ( f ) : Σ is T invariant subsystem } . ϕ : [0 , 1] → R , write S n ( ϕ )( x ) = ψ ( x ) + · · · + ψ ( T n − 1 x ) , var n ( ϕ ) := sup {| ϕ ( x ) − ϕ ( y ) | : x, y ∈ I ( a 1 , a 2 , · · · , a n ) } a 1 ,a 2 , ··· ,a n Tempered Distortion Property : 1 var 1 ( ϕ ) < ∞ and lim nvar n S n ( ϕ ) = 0 . n →∞ Jun Wu, Huazhong University of Sci. & Tech. The dimensional theory of continued fractions

Notation Generalizations Note that 1 1 a n +1 ( x ) + 1 ≤ T n x ≤ a n +1 ( x ) . So the growth rate of a n ( x ) can also be given by the speed that T n x approximate the point 0 . Shrinking target : For fixed y , how about the size of the set � � x : | T n x − y | ≤ ψ ( n, x ) , i.o. n ∈ N ? In particular, let f : [0 , 1] → R + , define x ∈ [0 , 1] : | T n x − y | ≤ e − S n f ( x ) , i.o. n � � S y ( f ) = , how about the size of S y ( f ) ? Jun Wu, Huazhong University of Sci. & Tech. The dimensional theory of continued fractions

Notation From the viewpoint of dynamical system Suppose we have a dynamical system ( X, T, µ ) , where µ is a T -invariant ergodic probability measure. Let A ⊂ X such that µ ( A ) > 0 . Ergodic property implies that ∞ ∞ � � T − n ( A )) = 1 , µ ( m =1 n = m that is, µ almost every x ∈ X will visit A an infinite number of times. This raises the question of what happens when we allow A to shrink with respect to time. How does the size of � ∞ � ∞ n = m T − n ( A ( n )) depend m =1 upon the sequence { A ( n ) } n ≥ 1 ? The shrinking target problem initialed by Hill and Velani (1995) which concerns “what happens if the target shrinks with the time and more generally if the target also moves around with the time.” Jun Wu, Huazhong University of Sci. & Tech. The dimensional theory of continued fractions