Hartman uniformly distributed sequences A sequence of integers ( a n ) n ≥ 1 is Hartman uniformly distributed if N 1 � lim e ( a n x ) = 0 N N →∞ n =1 for all non-integer x . Equivalenty a sequence is Hartmann uniformly distributed if ( { a n γ } ) n ≥ 1 is uniform distributed modulo 1 for each irrational number γ , and the sequence ( a n ) n ≥ 1 is uniformly distributed in each residue class mod m for each natural number m > 1. Note if n ∈ N then n 2 �≡ 3 mod 4 so in general the sequences ( φ ( n )) ∞ n =1 and ( φ ( p n )) ∞ n =1 are not Hartman uniformly distributed. We do however know that if β ∈ R \ Q then ( φ ( n ) β ) ∞ n =1 and ( φ ( p n ) β ) ∞ n =1 are uniformly distributed modulo one. Condition H sequences to follow are Hartman uniformly distributed.
Condition H sequences of integers 3. ( a n ) ∞ n =1 that are L p -good universal and Hartman uniformly distributed are constructed as follows.
Condition H sequences of integers 3. ( a n ) ∞ n =1 that are L p -good universal and Hartman uniformly distributed are constructed as follows. Denote by [ y ] the integer part of real number y
Condition H sequences of integers 3. ( a n ) ∞ n =1 that are L p -good universal and Hartman uniformly distributed are constructed as follows. Denote by [ y ] the integer part of real number y . Set a n = [ g ( n )] ( n = 1 , . . . ) where g : [1 , ∞ ) → [1 , ∞ ) is a differentiable function whose derivation increases with its argument.
Condition H sequences of integers 3. ( a n ) ∞ n =1 that are L p -good universal and Hartman uniformly distributed are constructed as follows. Denote by [ y ] the integer part of real number y . Set a n = [ g ( n )] ( n = 1 , . . . ) where g : [1 , ∞ ) → [1 , ∞ ) is a differentiable function whose derivation increases with its argument. Let A n denote the cardinality of the set { n : a n ≤ n } and suppose for some function a : [1 , ∞ ) → [1 , ∞ ) increasing to infinity as its argument does, that we set � � � � � � � b M = sup e ( za n ) . � � a ( M ) , 1 1 � � { z }∈ [ 2 ) n : a n ≤ M � �
Condition H sequences of integers 3. ( a n ) ∞ n =1 that are L p -good universal and Hartman uniformly distributed are constructed as follows. Denote by [ y ] the integer part of real number y . Set a n = [ g ( n )] ( n = 1 , . . . ) where g : [1 , ∞ ) → [1 , ∞ ) is a differentiable function whose derivation increases with its argument. Let A n denote the cardinality of the set { n : a n ≤ n } and suppose for some function a : [1 , ∞ ) → [1 , ∞ ) increasing to infinity as its argument does, � � that we set b M = sup { z }∈ [ �� n : a n ≤ M e ( za n ) � . Suppose also � � a ( M ) , 1 1 2 ) for some decreasing function c : [1 , ∞ ) → [1 , ∞ ), with � ∞ s =1 c ( θ s ) < ∞ for θ > 1 and some positive constant C > 0 that M b ( M ) + A [ a ( M )] + a ( M ) ≤ Cc ( M ) . A M
Condition H sequences of integers 3. ( a n ) ∞ n =1 that are L p -good universal and Hartman uniformly distributed are constructed as follows. Denote by [ y ] the integer part of real number y . Set a n = [ g ( n )] ( n = 1 , . . . ) where g : [1 , ∞ ) → [1 , ∞ ) is a differentiable function whose derivation increases with its argument. Let A n denote the cardinality of the set { n : a n ≤ n } and suppose for some function a : [1 , ∞ ) → [1 , ∞ ) increasing to infinity as its argument does, � � that we set b M = sup { z }∈ [ � n : a n ≤ M e ( za n ) � . Suppose also � � a ( M ) , 1 1 2 ) � for some decreasing function c : [1 , ∞ ) → [1 , ∞ ), with � ∞ s =1 c ( θ s ) < ∞ for θ > 1 and some positive constant C > 0 that M b ( M ) + A [ a ( M )] + a ( M ) ≤ Cc ( M ) . A M Then we say that k = ( a n ) ∞ n =1 satisfies condition H . (Nair)
Examples of Hartman uniformly distribution sequences Sequences satisfying condition H are both Hartman uniformly distributed and L p -good universal. Specific sequences of integers that satisfy conditions H include k n = [ g ( n )] ( n = 1 , 2 , . . . ) where I. g ( n ) = n ω if ω > 1 and ω / ∈ N . II. g ( n ) = e log γ n for γ ∈ (1 , 3 2 ). III. g ( n ) = P ( n ) = b k n k + . . . + b 1 n + b 0 for b k , . . . , b 1 not all rational multiplies of the same real number.
Bourgain’s random sequences 4. Suppose S = ( n k ) ∞ n =1 ⊆ N is a strictly increasing sequence of natural numbers. By identifying S with its characteristic function I S we may view it as a point in Λ = { 0 , 1 } N the set of maps from N to { 0 , 1 } .
Bourgain’s random sequences 4. Suppose S = ( n k ) ∞ n =1 ⊆ N is a strictly increasing sequence of natural numbers. By identifying S with its characteristic function I S we may view it as a point in Λ = { 0 , 1 } N the set of maps from N to { 0 , 1 } . We may endow Λ with a probability measure by viewing it as a Cartesian product Λ = � ∞ n =1 X n where for each natural number n we have X n = { 0 , 1 } and specify the probability π n on X n by π n ( { 1 } ) = q n with 0 ≤ q n ≤ 1 and π n ( { 0 } ) = 1 − q n such that lim n →∞ q n n = ∞ .
Bourgain’s random sequences 4. Suppose S = ( n k ) ∞ n =1 ⊆ N is a strictly increasing sequence of natural numbers. By identifying S with its characteristic function I S we may view it as a point in Λ = { 0 , 1 } N the set of maps from N to { 0 , 1 } . We may endow Λ with a probability measure by viewing it as a Cartesian product Λ = � ∞ n =1 X n where for each natural number n we have X n = { 0 , 1 } and specify the probability π n on X n by π n ( { 1 } ) = q n with 0 ≤ q n ≤ 1 and π n ( { 0 } ) = 1 − q n such that lim n →∞ q n n = ∞ . The desired probability measure on Λ is the corresponding product measure π = � ∞ n =1 π n . The underlying σ -algebra β is that generated by the “cylinders” { λ = ( λ n ) ∞ n =1 ∈ Λ : λ i 1 = α i 1 , . . . λ i r = α i r } for all possible choices of i 1 , . . . , i r and α i 1 , . . . , α i r .
Bourgain’s random sequences 4. Suppose S = ( n k ) ∞ n =1 ⊆ N is a strictly increasing sequence of natural numbers. By identifying S with its characteristic function I S we may view it as a point in Λ = { 0 , 1 } N the set of maps from N to { 0 , 1 } . We may endow Λ with a probability measure by viewing it as a Cartesian product Λ = � ∞ n =1 X n where for each natural number n we have X n = { 0 , 1 } and specify the probability π n on X n by π n ( { 1 } ) = q n with 0 ≤ q n ≤ 1 and π n ( { 0 } ) = 1 − q n such that lim n →∞ q n n = ∞ . The desired probability measure on Λ is the corresponding product measure π = � ∞ n =1 π n . The underlying σ -algebra β is that generated by the “cylinders” { λ = ( λ n ) ∞ n =1 ∈ Λ : λ i 1 = α i 1 , . . . λ i r = α i r } for all possible choices of i 1 , . . . , i r and α i 1 , . . . , α i r . Let ( k n ) ∞ n =1 be almost any point in Λ with respect to the measure π .
Means of convergents for subsequences Suppose the function F with domain the non-negative real numbers and range the real numbers is continuous and increasing. For each natural number n and arbitrary non-negative real numbers a 1 , · · · , a n we define n M F , n ( a 1 , · · · , a n ) = F − 1 [1 � F ( a j )] . n j =1 Then if ( a n ) n ≥ 1 is L p good universal and ( { a n γ } ) n ≥ 1 is uniformly distributed modulo one for irrational γ we have � 1 1 dt n →∞ M F , n ( c a 1 ( x ) , · · · , c a n ( x )) = F − 1 [ lim F ( c 1 ( t )) d 1 + t ] , log 2 − 0 almost every where with respect to Lebesgue measure.
Means of convergents for subsequences Suppose the function F with domain the non-negative real numbers and range the real numbers is continuous and increasing. For each natural number n and arbitrary non-negative real numbers � n a 1 , · · · , a n we define M F , n ( a 1 , · · · , a n ) = F − 1 [ 1 j =1 F ( a j )]. n Then if ( a n ) n ≥ 1 is L p good universal and ( { a n γ } ) n ≥ 1 is uniformly distributed modulo one for irrational γ we have � 1 1 dt n →∞ M F , n ( c a 1 ( x ) , · · · , c a n ( x )) = F − 1 [ lim F ( c 1 ( t )) d 1 + t ] , log 2 − 0 almost every where with respect to Lebesgue measure. Special cases (i) lim N →∞ 1 � N n =1 c a n ( x ) = ∞ a . e . ; N (ii) lim N →∞ ( c a 1 ( x ) . . . c a N ( x )) N − 1 = Π k ≥ 1 (1 + log k 1 log 2 a.e. k ( k +2) )
Hurwitz’s constants for subsequences Recall the inequality | x − p n 1 | ≤ , q 2 q n n which is classical and well known. Clearly n | x − p n θ n ( x ) = q 2 | ∈ [0 , 1) . q n if for each natural number n . Set � z 1 x ∈ [0 , 2 ); log 2 F ( x ) = 1 if x ∈ [ 1 log 2 (1 − z + log 2 z ) 2 , 1]
Hurwitz’s constants for subsequences p n 1 Recall the inequality | x − q n | ≤ n , which is classical and well q 2 p n known. Clearly θ n ( x ) = q 2 n | x − q n | ∈ [0 , 1) . if for each natural number n . Set � z 1 x ∈ [0 , 2 ); log 2 F ( x ) = 1 if x ∈ [ 1 log 2 (1 − z + log 2 z ) 2 , 1] Then if [ ( a n ) n ≥ 1 is L p good universal and ( { a n γ } ) n ≥ 1 is uniformly distributed modulo one for irrational γ ]* we have 1 lim n |{ 1 ≤ j ≤ n : θ a j ( x ) ≤ z }| = F ( z ) , n →∞ almost everywhere with respect to Lebesgue measure.
Hurwitz’s constants for subsequences p n 1 Recall the inequality | x − q n | ≤ n , which is classical and well q 2 p n known. Clearly θ n ( x ) = q 2 n | x − q n | ∈ [0 , 1) . if for each natural number n . Set � 1 z x ∈ [0 , 2 ); log 2 F ( x ) = 1 if x ∈ [ 1 log 2 (1 − z + log 2 z ) 2 , 1] Then if [ ( a n ) n ≥ 1 is L p good universal and ( { a n γ } ) n ≥ 1 is uniformly distributed modulo one for irrational γ ]* we have 1 lim n |{ 1 ≤ j ≤ n : θ a j ( x ) ≤ z }| = F ( z ) , n →∞ almost everywhere with respect to Lebesgue measure. D. Hensley dropped condition * using a different method.
Other sequences attached to the regular continued fraction expansion. I Suppose z is in [0 , 1] and for irrational x in (0 , 1) set Q n ( x ) = q n − 1 ( x ) q n ( x ) for each positive integer n . Suppose also that ( a n ) ∞ n =1 satisfies *. Then 1 n |{ 1 ≤ j ≤ n : Q a j ( x ) ≤ z }| = F 2 ( z ) = log(1 + z ) lim log 2 n →∞ almost everywhere with respect to Lebesgue measure.
Other sequences attached to the regular continued fraction expansion II For irrational x in (0 , 1) set p n | x − q n | ( n = 1 , 2 , · · · ) r n ( x ) = q n − 1 | . p n − 1 | x −
Other sequences attached to the regular continued fraction expansion II For irrational x in (0 , 1) set p n | x − q n | r n ( x ) = q n − 1 | . ( n = 1 , 2 , · · · ) p n − 1 | x − Further for z in [0 , 1] let 1 z F 3 ( z ) = log 2(log(1 + z ) − 1 + z log z ) .
Other sequences attached to the regular continued fraction expansion II For irrational x in (0 , 1) set p n | x − q n | r n ( x ) = q n − 1 | . ( n = 1 , 2 , · · · ) p n − 1 | x − Further for z in [0 , 1] let 1 z F 3 ( z ) = log 2(log(1 + z ) − 1 + z log z ) . Suppose also that ( a n ) ∞ n =1 satisfes *. Then 1 lim n |{ 1 ≤ j ≤ n : r a j ( x ) ≤ z }| = F 3 ( z ) , n →∞ almost everywhere with respect to Lebesgue measure.
Continued fraction map on [1, 0)
Markov Partitions Let T be a self map of [0 , 1] and let P 0 = { P ( j ) : j ∈ Λ } be a partition of [0 , 1] into open intervals, disregarding a set of Lebesgue measure 0.
Markov Partitions Let T be a self map of [0 , 1] and let P 0 = { P ( j ) : j ∈ Λ } be a partition of [0 , 1] into open intervals, disregarding a set of Lebesgue measure 0. We may for instance take Λ to be { 1 , 2 , · · · , n } ( n = 1 , 2 , · · · ) or we may take Λ to be the natural numbers.
Markov Partitions Let T be a self map of [0 , 1] and let P 0 = { P ( j ) : j ∈ Λ } be a partition of [0 , 1] into open intervals, disregarding a set of Lebesgue measure 0. We may for instance take Λ to be { 1 , 2 , · · · , n } ( n = 1 , 2 , · · · ) or we may take Λ to be the natural numbers. Define further partitions P k = { P ( j 0 , · · · , j k ) : ( j 0 , · · · , j k ) ∈ Λ k +1 } of [0 , 1] inductively for k in N by setting P ( j 0 , · · · , j k ) = P ( j 0 , · · · , j k − 1 ) ∩ T − k ( P ( j k )) , so that P k = P k − 1 ∨ T − 1 ( P k − 1 ) .
Markov Partitions Let T be a self map of [0 , 1] and let P 0 = { P ( j ) : j ∈ Λ } be a partition of [0 , 1] into open intervals, disregarding a set of Lebesgue measure 0. We may for instance take Λ to be { 1 , 2 , · · · , n } ( n = 1 , 2 , · · · ) or we may take Λ to be the natural numbers. Define further partitions P k = { P ( j 0 , · · · , j k ) : ( j 0 , · · · , j k ) ∈ Λ k +1 } of [0 , 1] inductively for k in N by setting P ( j 0 , · · · , j k ) = P ( j 0 , · · · , j k − 1 ) ∩ T − k ( P ( j k )) , so that P k = P k − 1 ∨ T − 1 ( P k − 1 ) . Of course some of these sets P ( j 0 , · · · , j k ) may be empty. We shall disregard these.
Markov Maps of the unit interval We say the map T is Markov with partition P 0 if the following conditions hold :
Markov Maps of the unit interval We say the map T is Markov with partition P 0 if the following conditions hold : (i) for each j in Λ there exists Λ j ⊂ Λ such that T ( P ( j )) = int ∪ i ∈ Λ j P ( i );
Markov Maps of the unit interval We say the map T is Markov with partition P 0 if the following conditions hold : (i) for each j in Λ there exists Λ j ⊂ Λ such that T ( P ( j )) = int ∪ i ∈ Λ j P ( i ); (ii) we have inf j ∈ Λ λ ( T ( P ( j )) > 0;
Markov Maps of the unit interval We say the map T is Markov with partition P 0 if the following conditions hold : (i) for each j in Λ there exists Λ j ⊂ Λ such that T ( P ( j )) = int ∪ i ∈ Λ j P ( i ); (ii) we have inf j ∈ Λ λ ( T ( P ( j )) > 0; (iii) the derivative T ′ of T is defined and 1 T ′ is bounded off endpoints;
Markov Maps of the unit interval We say the map T is Markov with partition P 0 if the following conditions hold : (i) for each j in Λ there exists Λ j ⊂ Λ such that T ( P ( j )) = int ∪ i ∈ Λ j P ( i ); (ii) we have inf j ∈ Λ λ ( T ( P ( j )) > 0; (iii) the derivative T ′ of T is defined and 1 T ′ is bounded off endpoints;
Markov Maps of the unit interval We say the map T is Markov with partition P 0 if the following conditions hold : (i) for each j in Λ there exists Λ j ⊂ Λ such that T ( P ( j )) = int ∪ i ∈ Λ j P ( i ); (ii) we have inf j ∈ Λ λ ( T ( P ( j )) > 0; (iii) the derivative T ′ of T is defined and 1 T ′ is bounded on U 0 ; (iv) there exists β > 1 such that ( T n ) ′ ≫ β n off endpoints;
Markov Maps of the unit interval We say the map T is Markov with partition P 0 if the following conditions hold : (i) for each j in Λ there exists Λ j ⊂ Λ such that T ( P ( j )) = int ∪ i ∈ Λ j P ( i ); (ii) we have inf j ∈ Λ λ ( T ( P ( j )) > 0; (iii) the derivative T ′ of T is defined and 1 T ′ is bounded on U 0 ; (iv) there exists β > 1 such that ( T n ) ′ ≫ β n on U n and (v) there exists γ in (0 , 1) such that | 1 − T ′ ( x ) T ′ ( y ) | ≪ | x − y | γ , for x and y belonging to the same element of P 0 .
Examples of Markov Maps (a) For a Pisot-Vijayaraghavan number β > 1 let T β ( x ) = { β x } ;
Examples of Markov Maps (a) For a Pisot-Vijayaraghavan number β > 1 let T β ( x ) = { β x } ; and let (b) �� 1 � if x � = 0; x Tx = 0 if x = 0 ,
Examples of Markov Maps (a) For a Pisot-Vijayaraghavan number β > 1 let T β ( x ) = { β x } ; and let (b) �� 1 � if x � = 0; x Tx = 0 if x = 0 , The example (a) is known as the β -transformation. Note that in the special case where β is an integer T β ( T β ( x )) = { β 2 x } . This is not true for non-integer β and this gives the dynamics a quite different character. The example (b) is the famous Gauss map which is associated to the continued fraction expansion of a real number.
Continued fraction map on [1, 0)
Invariant Measures for Markov Measures If the map T : [0 , 1] → [0 , 1] is Markov in the sense described above then it preserves a measure η equivalent to Lebesgue measure. Further the dynamical system ([0 , 1] , β, η, T ), where β denotes the usual Borel σ -algebra on [0 , 1], is exact. In particular it is ergodic.
Invariant Measures for Markov Measures If the map T : [0 , 1] → [0 , 1] is Markov in the sense described above then it preserves a measure η equivalent to Lebesgue measure. Further the dynamical system ([0 , 1] , β, η, T ), where β denotes the usual Borel σ -algebra on [0 , 1], is exact. In particular it is ergodic. As a consequence, G. Birkhoff’s pointwise ergodic theorem tells us that N 1 � χ B ( T n ( x )) = η ( B ) , (1 . 1) lim N N →∞ n =1 almost everywhere with respect to Lebesgue measure.
An exceptional set For x in [0 , 1] let Ω( x ) = Ω T ( x ) denote the closure of the set { T n ( x ) : n = 1 , 2 , · · · } .
An exceptional set For x in [0 , 1] let Ω( x ) = Ω T ( x ) denote the closure of the set { T n ( x ) : n = 1 , 2 , · · · } . Henceforth we denote N ∪ { 0 } by N 0 .
An exceptional set For x in [0 , 1] let Ω( x ) = Ω T ( x ) denote the closure of the set { T n ( x ) : n = 1 , 2 , · · · } . Henceforth we denote N ∪ { 0 } by N 0 . For a subset A of [0 , 1] let d ( x , A ) denote inf a ∈ A | x − a | .
An exceptional set For x in [0 , 1] let Ω( x ) = Ω T ( x ) denote the closure of the set { T n ( x ) : n = 1 , 2 , · · · } . Henceforth we denote N ∪ { 0 } by N 0 . For a subset A of [0 , 1] let d ( x , A ) denote inf a ∈ A | x − a | . If x = ( x r ) ∞ r =0 is a sequence of real numbers such that 0 ≤ x r ≤ 1 and f : N 0 → R is positive, set E ( x , f ) = { x ∈ [0 , 1] : | log d ( x r , Ω( x )) | ≪ f ( r ) } .
An exceptional set For x in [0 , 1] let Ω( x ) = Ω T ( x ) denote the closure of the set { T n ( x ) : n = 1 , 2 , · · · } . Henceforth we denote N ∪ { 0 } by N 0 . For a subset A of [0 , 1] let d ( x , A ) denote inf a ∈ A | x − a | . If x = ( x r ) ∞ r =0 is a sequence of real numbers such that 0 ≤ x r ≤ 1 and f : N 0 → R is positive, set E ( x , f ) = { x ∈ [0 , 1] : | log d ( x r , Ω( x )) | ≪ f ( r ) } . As a consequence of Birkhoff’s theorem and the fact that η is equivalent to Lebesgue measure λ we see that λ ( E ( x , f )) = 0.
Hausdorff Dimension Suppose M is a metric space endowed with a metric d .
Hausdorff Dimension Suppose M is a metric space endowed with a metric d . Also suppose E ⊆ M .
Hausdorff Dimension Suppose M is a metric space endowed with a metric d . Also suppose E ⊆ M . Suppose δ > 0.
Hausdorff Dimension Suppose M is a metric space endowed with a metric d . Also suppose E ⊆ M . Suppose δ > 0. We say a collection of subsets of M denoted C δ is a δ –cover for E if E ⊆ ∪ U ∈C δ U , and if we set diam ( U ) := sup d ( x , y ) , x , y ∈ U then U ∈ C δ implies diam ( U ) ≤ δ .
Hausdorff Dimension Suppose M is a metric space endowed with a metric d . Also suppose E ⊆ M . Suppose δ > 0. We say a collection of subsets of M denoted C δ is a δ –cover for E if E ⊆ ∪ U ∈C δ U , and if we set diam ( U ) := sup d ( x , y ) , x , y ∈ U then U ∈ C δ implies diam ( U ) ≤ δ . We set H s � ( diamU i ) s , δ ( E ) = sup C δ i where the supremum is taken over all δ -covers C δ .
Hausdorff Dimension Suppose M is a metric space endowed with a metric d . Also suppose E ⊆ M . Suppose δ > 0. We say a collection of subsets of M denoted C δ is a δ –cover for E if E ⊆ ∪ U ∈C δ U , and if we set diam ( U ) := sup d ( x , y ) , x , y ∈ U then U ∈ C δ implies diam ( U ) ≤ δ . We set � H s ( diamU i ) s , δ ( E ) = sup C δ i where the supremum is taken over all δ -covers C δ . We set H s ( E ) := lim δ → 0 H s δ ( E ) , which always exists.
Hausdorff Dimension Suppose M is a metric space endowed with a metric d . Also suppose E ⊆ M . Suppose δ > 0. We say a collection of subsets of M denoted C δ is a δ –cover for E if E ⊆ ∪ U ∈C δ U , and if we set diam ( U ) := sup d ( x , y ) , x , y ∈ U then U ∈ C δ implies diam ( U ) ≤ δ . We set � H s ( diamU i ) s , δ ( E ) = sup C δ i where the supremum is taken over all δ -covers C δ . We set H s ( E ) := lim δ → 0 H s δ ( E ) , which always exists. We call the specific s 0 where H s changes from ∞ to 0 the Hausdorff dimension of E .
Some examples (i) If M = R n for n > 1 and E ⊆ M has positive lebesgue measure then s 0 = n i.e. dim ( E ) = n .
Some examples (i) If M = R n for n > 1 and E ⊆ M has positive lebesgue measure then s 0 = n i.e. dim ( E ) = n . (ii) If E ⊆ M is countable then dim ( E ) = 0.
Some examples (i) If M = R n for n > 1 and E ⊆ M has positive lebesgue measure then s 0 = n i.e. dim ( E ) = n . (ii) If E ⊆ M is countable then dim ( E ) = 0. (iii) Cantor’s middle third set : Let ∞ x n � C = { x ∈ [0 , 1) : x = 3 n s . t . x n ∈ { 0 , 2 }} . n =1 C is well known to be uncountanle. One can show dim ( C ) = log 2 log 3 .
Abercrombie, Nair For each sequence x = ( x r ) ∞ r =0 of real numbers in [0 , 1] and positive function f : N 0 → R such that f ( r ) ≫ r 2 , the Hausdorff dimension of E ( x , f ) is 1.
A special case An immediate consequence is the following result. For x 0 ∈ [0 , 1] set E ( x 0 ) = { x ∈ [0 , 1] : x 0 ∈ [0 , 1] \ Ω T ( x ) } . Then for each x 0 in [0 , 1] the Hausdorff dimension of E ( x 0 ) is 1.
A special case An immediate consequence is the following result. For x 0 ∈ [0 , 1] set E ( x 0 ) = { x ∈ [0 , 1] : x 0 ∈ [0 , 1] \ Ω T ( x ) } . Then for each x 0 in [0 , 1] the Hausdorff dimension of E ( x 0 ) is 1. Take x 0 = 0 and T is the Gauss continued fraction map. Thus the set of x ∈ [0 , 1] with bound convergents had dimension 1.
Continued fraction map on [1, 0)
Equivalent characterisations of badly approximability (i) We say an irrational real number α is badly approximable if p c ( α ) there exists a constant c ( α ) > 0 such that | α − q | > q 2 , for every rational p q .
Equivalent characterisations of badly approximability (i) We say an irrational real number α is badly approximable if c ( α ) p there exists a constant c ( α ) > 0 such that | α − q | > q 2 , for every rational p q . (ii) Suppose α has a continued fraction expansion [ a 0 ; a 1 , a 2 , . . . ]. We say α has bounded partial quotients if there exists a constant K ( α ) such that | c n | ≤ K ( α ) . ( n = 1 , 2 , · · · )
Equivalent characterisations of badly approximability (i) We say an irrational real number α is badly approximable if c ( α ) p there exists a constant c ( α ) > 0 such that | α − q | > q 2 , for every rational p q . (ii) Suppose α has a continued fraction expansion [ a 0 ; a 1 , a 2 , . . . ]. We say α has bounded partial quotients if there exists a constant K ( α ) such that | c n | ≤ K ( α ) . ( n = 1 , 2 , · · · ) (i) and (ii) are equivalent. Corollary : (V. Jarnik 1929) : The set of badly approximable numbers has Hausdorff dimension 1
The Field of Formal Power series Let F q denote the finite field of q elements, where q is a power of a prime p . If Z is an indeterminate, we denote by F q [ Z ] and F q ( Z ) the ring of polynomials in Z with coefficients in F q and the quotient field of F q [ Z ] , respectively.
The Field of Formal Power series Let F q denote the finite field of q elements, where q is a power of a prime p . If Z is an indeterminate, we denote by F q [ Z ] and F q ( Z ) the ring of polynomials in Z with coefficients in F q and the quotient field of F q [ Z ] , respectively. For each P , Q ∈ F q [ Z ] with Q � = 0 , define | P / Q | = q deg( P ) − deg( Q ) and | 0 | = 0.
The Field of Formal Power series Let F q denote the finite field of q elements, where q is a power of a prime p . If Z is an indeterminate, we denote by F q [ Z ] and F q ( Z ) the ring of polynomials in Z with coefficients in F q and the quotient field of F q [ Z ] , respectively. For each P , Q ∈ F q [ Z ] with Q � = 0 , define | P / Q | = q deg( P ) − deg( Q ) and | 0 | = 0 . The field F q (( Z − 1 )) of formal Laurent series is the completion of F q ( Z ) with respect to the valuation | · | .
The Field of Formal Power series Let F q denote the finite field of q elements, where q is a power of a prime p . If Z is an indeterminate, we denote by F q [ Z ] and F q ( Z ) the ring of polynomials in Z with coefficients in F q and the quotient field of F q [ Z ] , respectively. For each P , Q ∈ F q [ Z ] with Q � = 0 , define | P / Q | = q deg( P ) − deg( Q ) and | 0 | = 0 . The field F q (( Z − 1 )) of formal Laurent series is the completion of F q ( Z ) with respect to the valuation | · | . That is, F q (( Z − 1 )) = { a n Z n + · · · + a 0 + a − 1 Z − 1 + · · · : n ∈ Z , a i ∈ F q } and we have | a n Z n + a n − 1 Z n − 1 + · · · | = q n ( a n � = 0) and | 0 | = 0 , where q is the number of elements of F q .
Haar measure on the field of Formal Power Series It is worth keeping in mind that | · | is a non-Archimedean norm, since | α + β | ≤ max( | α | , | β | ) . In fact, F q (( Z − 1 )) is the non-Archimedean local field of positive characteristic p .
Haar measure on the field of Formal Power Series It is worth keeping in mind that | · | is a non-Archimedean norm, since | α + β | ≤ max( | α | , | β | ) . In fact, F q (( Z − 1 )) is the non-Archimedean local field of positive characteristic p . As a result, there exists a unique, up to a positive multiplicative constant, countably additive Haar measure µ q on the Borel subsets of F q (( Z − 1 )) .
Haar measure on the field of Formal Power Series It is worth keeping in mind that | · | is a non-Archimedean norm, since | α + β | ≤ max( | α | , | β | ) . In fact, F q (( Z − 1 )) is the non-Archimedean local field of positive characteristic p . As a result, there exists a unique, up to a positive multiplicative constant, countably additive Haar measure µ q on the Borel subsets of F q (( Z − 1 )) . zuk found a characterization of Haar measure on F q (( Z − 1 )) Sprindˇ by its value on the balls B ( α ; q n ) = { β ∈ F q (( Z − 1 )): | α − β | < q n } .
Haar measure on the field of Formal Power Series It is worth keeping in mind that | · | is a non-Archimedean norm, since | α + β | ≤ max( | α | , | β | ) . In fact, F q (( Z − 1 )) is the non-Archimedean local field of positive characteristic p . As a result, there exists a unique, up to a positive multiplicative constant, countably additive Haar measure µ q on the Borel subsets of F q (( Z − 1 )) . zuk found a characterization of Haar measure on F q (( Z − 1 )) Sprindˇ by its value on the balls B ( α ; q n ) = { β ∈ F q (( Z − 1 )): | α − β | < q n } . It was shown that the equation µ q ( B ( α ; q n )) = q n completely characterizes Haar measure here.
Continued fractions on F q (( Z − 1 )) For each α ∈ F q (( Z − 1 )) , we can uniquely write 1 α = A 0 + = [ A 0 ; A 1 , A 2 , . . . ] , 1 A 1 + A 2 + ... where ( A n ) ∞ n =0 is a sequence of polynomials in F q [ Z ] with | A n | > 1 for all n ≥ 1 .
Continued fractions on F q (( Z − 1 )) For each α ∈ F q (( Z − 1 )) , we can uniquely write 1 α = A 0 + = [ A 0 ; A 1 , A 2 , . . . ] , 1 A 1 + A 2 + ... where ( A n ) ∞ n =0 is a sequence of polynomials in F q [ Z ] with | A n | > 1 for all n ≥ 1 . We define recursively the two sequences of polynomials ( P n ) ∞ n =0 and ( Q n ) ∞ n =0 by P n = A n P n − 1 + P n − 2 and Q n = A n Q n − 1 + Q n − 2 , with the initial conditions P 0 = A 0 , Q 0 = 1 , P 1 = A 1 A 0 + 1 and Q 1 = A 1 .
Continued fractions on F q (( Z − 1 )) For each α ∈ F q (( Z − 1 )) , we can uniquely write 1 α = A 0 + = [ A 0 ; A 1 , A 2 , . . . ] , 1 A 1 + A 2 + ... where ( A n ) ∞ n =0 is a sequence of polynomials in F q [ Z ] with | A n | > 1 for all n ≥ 1 . We define recursively the two sequences of polynomials ( P n ) ∞ n =0 and ( Q n ) ∞ n =0 by P n = A n P n − 1 + P n − 2 and Q n = A n Q n − 1 + Q n − 2 , with the initial conditions P 0 = A 0 , Q 0 = 1 , P 1 = A 1 A 0 + 1 and Q 1 = A 1 . Then we have Q n P n − 1 − P n Q n − 1 = ( − 1) n , and whence P n and Q n are coprime. In addition, we have P n / Q n = [ A 0 ; A 1 , . . . , A n ].
Continued fractions map on F q (( Z − 1 )) Define T q on the unit ball B (0; 1) = { a − 1 Z − 1 + a − 2 Z − 2 + · · · : a i ∈ F q } by � 1 � T q α = and T 0 = 0 . α
Continued fractions map on F q (( Z − 1 )) Define T q on the unit ball B (0; 1) = { a − 1 Z − 1 + a − 2 Z − 2 + · · · : a i ∈ F q } by � 1 � T q α = and T 0 = 0 . α Here { a n Z n + · · · + a 0 + a − 1 Z − 1 + · · · } = a − 1 Z − 1 + a − 2 Z − 2 + · · · denotes its fractional part.
Continued fractions map on F q (( Z − 1 )) Define T q on the unit ball B (0; 1) = { a − 1 Z − 1 + a − 2 Z − 2 + · · · : a i ∈ F q } by � 1 � T q α = and T 0 = 0 . α Here { a n Z n + · · · + a 0 + a − 1 Z − 1 + · · · } = a − 1 Z − 1 + a − 2 Z − 2 + · · · denotes its fractional part. We note that if α = [0; A 1 ( α ) , A 2 ( α ) , . . . ] , then we have, for all m , n ≥ 1 , T n α = [0; A n +1 ( α ) , A n +2 ( α ) , . . . ] A m ( T n α ) = A n + m ( α ) . and
Exactness the CF map on F q (( Z − 1 )) Let ( X , B , µ, T ) be a dynamical system consisting of a set X with the σ -algebra B of its subsets, a probability measure µ, and a transformation T : X → X . We say that ( X , B , µ, T ) is measure-preserving if, for all E ∈ B , µ ( T − 1 E ) = µ ( E ) .
Recommend
More recommend
