On the frequency of partial quotients of regular continued fractions
aa r X i v : . [ m a t h . D S ] J un ON THE FREQUENCY OF PARTIAL QUOTIENTS OFREGULAR CONTINUED FRACTIONS
AI-HUA FAN, LING-MIN LIAO, AND JI-HUA MA
Abstract.
We consider sets of real numbers in [0 ,
1) with prescribed frequen-cies of partial quotients in their regular continued fraction expansions. It isshown that the Hausdorff dimensions of these sets, always bounded from belowby 1 /
2, are given by a modified variational principle. Introduction
Let Q c denote the set of irrational number. It is well-known that each x ∈ [0 , ∩ Q c possesses a unique continued fraction expansion of the form x = 1 a ( x ) + 1 a ( x ) + 1 a ( x ) + . . . , (1.1)where a k ( x ) ∈ N := { , , , · · · } is the k -th partial quotient of x . This expansion isusually denoted by x = [ a ( x ) , a ( x ) , · · · ]. For each j ∈ N , define the frequency ofthe digit j in the continued fraction expansion of x by τ j ( x ) := lim n →∞ τ j ( x, n ) n , whenever the limit exists, where τ j ( x, n ) := Card { ≤ k ≤ n : a k ( x ) = j } . This paper is concerned with sets of real numbers with prescribed digit frequen-cies in their continued fraction expansions. To be precise, let ~p = ( p , p , . . . ) bea probability vector with p j ≥ j ∈ N and P ∞ j =1 p j = 1, which will becalled a frequency vector in the sequel. Our purpose is to determine the Hausdorffdimension of the set E ~p := { x ∈ [0 , ∩ Q c : τ j ( x ) = p j ∀ j ≥ } . Let us first recall some notation. For any a , a , · · · , a n ∈ N , we call I ( a , a , · · · , a n ) := { x ∈ [0 ,
1) : a ( x ) = a , a ( x ) = a , · · · , a n ( x ) = a n } a rank- n basic interval . Let T : [0 , → [0 ,
1) be the Gauss transformation definedby T (0) = 0 , T ( x ) = 1 /x (mod 1) for x ∈ (0 , . For a given frequency vector ~p = ( p , p , . . . ), we denote by N ( ~p ) the set of T -invariant ergodic probability measures µ such that Z | log x | dµ < ∞ and µ ( I ( j )) = p j for all j ≥ . (1.2) Let h µ stand for the measure-theoretical entropy of µ , and dim H A for the Hausdorffdimension of a set A . The main result of this paper can be stated as follows. Theorem 1.1.
For any frequency vector ~p , one has dim H ( E ~p ) = max ( , sup µ ∈N ( ~p ) h µ R | log x | dµ ) , where the “sup” is set to be zero if N ( ~p ) = ∅ .By virtue of log | T ′ ( x ) | = 2 | log x | , we see that 2 R | log x | dµ is the Liapunov ex-ponent of the measure µ . Therefore, the “sup” in the above is a variational formulawhich relates the Hausdorff dimension to the entropy and Liapunov exponent ofmeasures.Theorem 1.1 provides a complete solution to the long standing problem thatrequests an exact formula for dim H ( E ~p ). Let us recall some partial results in theliterature. In 1966, under the condition that P ∞ j =1 p j log j < ∞ , Kinney and Pitcher[9] showed that dim H ( E ~p ) ≥ − P ∞ j =1 p j log p j R | log x | dµ ~p , where µ ~p is the Bernoulli measure on [0 ,
1] defined by µ ~p ( I ( a , a , · · · , a n )) = n Y j =1 p a j . The above lower bound is just the Hausdorff dimension of the Bernoulli measure µ ~p . However, by a result of Kifer, Peres and Weiss in 2001, this is not an optimallower bound. Indeed, it is shown in [8] that, for any Bernoulli measure µ ~p ,dim H µ ~p ≤ − − This surprising fact indicates that the collection of Bernoulli measures are insuffi-cient for providing the correct lower bound for dim H ( E ~p ).In 1975, under the same condition P ∞ j =1 p j log j < ∞ , Billingsley and Hen-ningsen [2] obtained an improved lower bounddim H ( E ~p ) ≥ sup µ ∈N ( ~p ) h µ R | log x | dµ . Moreover, they proved that, for any fixed N ∈ N , this lower bound is the exactHausdorff dimension of the set { x ∈ E ~p : a n ( x ) ≤ N for all n ≥ } provided that p j = 0 for all j > N . It is therefore quite natural to guess that thislower bound is the right value for dim H E ~p in general. However, as will be shownin Theorem 1.1, this is not the case.Actually, the lower bound due to Billingsley and Henningsen is only a half of thecorrect lower bound. The other half of the lower bound, namely, dim H ( E ~p ) ≥ / N THE FREQUENCY OF PARTIAL QUOTIENTS OF REGULAR CONTINUED FRACTIONS3
The paper is organized as follows. In Section 2, we give some preliminaries onthe basic intervals and on the entropy of finite words. In Section 3, we establish theupper bound in Theorem 1.1. In Section 4, we prove that dim H ( E ~p ) ≥ / P ∞ j =1 p j log j < ∞ in Billingsley and Henningsen’stheorem and then obtain the lower bound in Theorem 1.1. The last section servesas a remark. 2. Preliminary
Let x = [ a ( x ) , a ( x ) , · · · ] ∈ [0 , ∩ Q c . The n -th convergent in the continuedfraction expansion of x is defined by p n q n := p n ( a ( x ) , · · · , a n ( x )) q n ( a ( x ) , · · · , a n ( x )) = 1 a ( x ) + 1 a ( x ) + . . . + 1 a n ( x ) . For ease of notation, we shall drop the argument x in what follows. It is known(see [6] p.9) that p n , q n can be obtained by the recursive relations: p − = 1 , p = 0 , p n = a n p n − + p n − ( n ≥ ,q − = 0 , q = 1 , q n = a n q n − + q n − ( n ≥ . By the above recursion relations, we have the following results.
Lemma 2.1 ([6]) . Let q n = q n ( a , · · · , a n ) and p n = p n ( a , · · · , a n ) , we have (i) p n − q n − p n q n − = ( − n ; (ii) q n ≥ n − , n Q k =1 a k ≤ q n ≤ n Q k =1 ( a k + 1) . Lemma 2.2 ([14]) . For any a , a , · · · , a n , b ∈ N , b + 12 ≤ q n +1 ( a , · · · , a j , b, a j +1 , · · · , a n ) q n ( a , · · · , a j , a j +1 , · · · , a n ) ≤ b + 1 ( ∀ ≤ j < n ) . Recall that for any a , a , · · · , a n ∈ N , the set I ( a , a , · · · , a n ) = { x ∈ [0 ,
1) : a ( x ) = a , a ( x ) = a , · · · , a n ( x ) = a n } is a rank- n basic interval. We write | I | for the length of an interval I . Lemma 2.3 ([10] p.18) . The basic interval I ( a , a , · · · , a n ) is an interval withendpoints p n /q n and ( p n + p n − ) / ( q n + q n − ) . Consequently, one has (cid:12)(cid:12)(cid:12) I ( a , · · · , a n ) (cid:12)(cid:12)(cid:12) = 1 q n ( q n + q n − ) , (2.1) and q n ≤ (cid:12)(cid:12)(cid:12) I ( a , · · · , a n ) (cid:12)(cid:12)(cid:12) ≤ q n . (2.2) Lemma 2.4.
We have | I ( x , · · · , j, · · · , x n ) | ≤ j + 1) (cid:12)(cid:12)(cid:12) I ( x , · · · , b j, · · · , x n ) (cid:12)(cid:12)(cid:12) , where the notation b j means “deleting the digit j ”. AI-HUA FAN, LING-MIN LIAO, AND JI-HUA MA
Proof.
By Lemma 2.2 and (2.2), we have | I ( x , · · · , j, · · · , x n ) | ≤ q − n ( x , · · · , j, · · · , x n ) ≤ (cid:18) j + 1 (cid:19) q − n − ( x , · · · , ˆ j, · · · , x n ) ≤ j + 1) (cid:12)(cid:12)(cid:12) I ( x , · · · , b j, · · · , x n ) (cid:12)(cid:12)(cid:12) . (cid:3) We will simply denote by I n ( x ) the rank n basic interval containing x . Supposethat a n := a n ( x ) ≥ I ′ n ( x ) = I ( a , · · · , a n − , a n −
1) and I ′′ n ( x ) = I ( a , · · · , a n − , a n + 1)which are two rank n basic intervals adjacent to I n ( x ). By the recursive equationof q n and (2.1), one has the following lemma. Lemma 2.5.
Suppose that a n := a n ( x ) ≥ . Then the lengths of the adjacentintervals I ′ n ( x ) and I ′′ n ( x ) are bounded by (cid:12)(cid:12) I n ( x ) (cid:12)(cid:12) from below and by (cid:12)(cid:12) I n ( x ) (cid:12)(cid:12) fromabove. For any x ∈ [0 , \ Q and any word i · · · i k ∈ N k , ( k ≥ τ i ··· i k ( x, n )the number of j , 1 ≤ j ≤ n , for which a j ( x ) · · · a j + k − ( x ) = i · · · i k . For N ∈ N , define Σ N := { , . . . , N } . We shall use the following estimate in [2]. Lemma 2.6 ([2]) . Let N ≥ and n ≥ . For any x = [ x , x , · · · ] ∈ [0 , ∩ Q c with x j ∈ Σ N for ≤ j ≤ n . Then for any k ≥ , we have log | I n ( x ) | ≤ X i ··· i k ∈ Σ kN τ i ··· i k ( x, n ) log p k ( i , · · · , i k ) q k ( i , · · · , i k ) + 8 + 8 n k . (2.3)Now we turn to the key combinatorial lemma which will be used in the upperbound estimation. Let φ : [0 , → R denote the function φ (0) = 0 , and φ ( t ) = − t log t for 0 < t ≤ . For every word ω ∈ Σ nN of length n and every word u ∈ Σ kN of length k , denote by p ( u | ω ) the frequency of appearances of u in ω , i.e., p ( u | ω ) = τ u ( ω ) n − k + 1 , where τ u ( ω ) denote the number of j , 1 ≤ j ≤ n − k + 1, for which ω j · · · ω j + k − = u. Define H k ( ω ) := X u ∈ Σ kN φ ( p ( u | ω )) . We have the following counting lemma.
Lemma 2.7 ([5]) . For any h > , ǫ > , k ∈ N , and for any n ∈ N large enough,we have Card { ω ∈ Σ nN : H k ( ω ) ≤ kh } ≤ exp( n ( h + ǫ )) . N THE FREQUENCY OF PARTIAL QUOTIENTS OF REGULAR CONTINUED FRACTIONS5 Upper bound
Some Lemmas.
Let ( p ( i , · · · , i k )) i ··· i k ∈ N k be a probability vector indexedby N k . As usual, we denote by q k ( a , · · · , a k ) the denominator of the k -th conver-gent of a real number with leading continued fraction digits a , . . . , a k . Lemma 3.1.
For each k ∈ N and each probability vector ( p ( i , · · · , i k )) i ··· i k ∈ N k , X i ··· i k ∈ N k − p ( i , · · · , i k ) log p ( i , · · · , i k ) ≤ X i ··· i k ∈ N k − p ( i , · · · , i k ) log | I ( i , · · · , i k ) | . Proof.
Applying Jesen’s inequality to the concave function log, we have X i ··· i k ∈ N k p ( i , · · · , i k ) log | I ( i , · · · , i k ) | p ( i , · · · , i k ) ≤ log X i ··· i k ∈ N k | I ( i , · · · , i k ) | = 0 . (cid:3) Lemma 3.2.
Let ~p = ( p , p , . . . ) be a probability vector and ~q = ( q , q , . . . ) apositive vector. Suppose − P ∞ j =1 p j log q j = ∞ and P ∞ j =1 q sj < ∞ for some positivenumber s . Then lim sup n →∞ − P nj =1 p j log p j − P nj =1 p j log q j ≤ s. Proof.
This is a consequence of the following inequality (see [13], p.217): for non-negative numbers s j (1 ≤ j ≤ m ) such that P mj =1 s j = 1 and any real numbers t j (1 ≤ j ≤ m ), we have m X j =1 s j ( t j − log s j ) ≤ log( m X j =1 e t j ) . (3.1)Fix n ≥
1. Let s j = p j for 1 ≤ j ≤ n and s n +1 = P ∞ j = n +1 p j . Let t j = s log q j for 1 ≤ j ≤ n and t n +1 = 0. Applying the above inequality (3.1) with m = n + 1,we get s n X j =1 p j log q j − n X j =1 p j log p j − ( ∞ X j = n +1 p j ) log( ∞ X j = n +1 p j ) ≤ log(1 + n X j =1 q sj ) . Consequently, − P nj =1 p j log p j − P nj =1 p j log q j ≤ s + ( P ∞ j = n +1 p j ) log( P ∞ j = n +1 p j ) − P nj =1 p j log q j + log(1 + P nj =1 q sj ) − P nj =1 p j log q j . Using the facts − P ∞ j =1 p j log q j = ∞ and P ∞ j =1 q sj < ∞ , we finish the proof byletting n → ∞ . (cid:3) Lemma 3.2 implies the following lemma. Recall that Σ kN = { , . . . , N } k . Lemma 3.3.
Let k ≥ . If ( p ( i , · · · , i k )) i ··· i k ∈ N k is a probability vector such that X i ··· i k ∈ N k p ( i , · · · , i k ) log q k ( i , · · · , i k ) = ∞ , then we have lim sup N →∞ − P i ··· i k ∈ Σ kN p ( i , · · · , i k ) log p ( i , · · · , i k )2 P i ··· i k ∈ Σ kN p ( i , · · · , i k ) log q k ( i , · · · , i k ) ≤ . AI-HUA FAN, LING-MIN LIAO, AND JI-HUA MA
Proof.
By Lemma 2.1, for any k ∈ N and any s > /
2, we have X i , ··· ,i k q k ( i , · · · , i k ) − s ≤ X i , ··· ,i k ( i · · · i k ) − s = ( ∞ X j =1 j − s ) k < ∞ . Thus we get the result by Lemma 3.2. (cid:3)
Proof of the upper bound.
To prove the upper bound, we shall make useof multi-step Markov measures. Let k ≥
1, by a ( k − T -invariant probability measure P on [0 ,
1) satisfying the Markov property P ( I ( a , · · · , a n )) P ( I ( a , · · · , a n − )) = P ( I ( a n − k , · · · , a n )) P ( I ( a n − k , · · · , a n − )) (3.2)for all n ≥ a , · · · , a n ∈ N (see [3], p.9). We may regard a Bernoulli measureas a 0-step Markov measure.For each N ≥
2, we denote by P kN = P kN ( ~p ) the collection of ( k − P ( I ( j )) = p j for 1 ≤ j ≤ N − P ( I ( N )) = 1 − N − X j =1 p j . (3.3)These Markov measures are supported by the set of continued fractions for whichthe partial quotients are bounded from above by N .For each i · · · i k ∈ { , , · · · , N } k , write p ( i , . . . , i k ) = P ( I ( i , . . . , i k )). Put α N,k := sup P ∈ P kN − k P p ( i , · · · , i k ) log p ( i , · · · , i k ) − P p ( i , · · · , i k ) log( p k ( i , · · · , i k ) /q k ( i , · · · , i k )) . (3.4)The argument in [2] (pp.171-172) shows that the following limit α N := lim k →∞ α N,k , exists and coincides with each of the following three limits: α ′ N := lim k →∞ sup P ∈ P kN − P p ( i , · · · , i k ) log p ( i , · · · , i k )2 P p ( i , · · · , i k ) log q k ( i , · · · , i k ) ,α ′′ N := lim k →∞ sup P ∈ P kN − P p ( i , · · · , i k ) log p ( i , · · · , i k ) − P p ( i , · · · , i k ) log | I ( i , · · · , i k ) | , and α ′′′ N := lim k →∞ sup P ∈ P kN h P R | log x | dP . Let α := lim sup N →∞ α N = lim sup N →∞ α ′ N = lim sup N →∞ α ′′ N = lim sup N →∞ α ′′′ N . (3.5)To prove the upper bound, we need only to prove the following two propositions. Proposition 3.4.
For any N ∈ N large enough, we have dim H ( E ~p ) ≤ max (cid:26) , α N (cid:27) . N THE FREQUENCY OF PARTIAL QUOTIENTS OF REGULAR CONTINUED FRACTIONS7
Proposition 3.5.
We have α ≤ max ( , sup µ ∈N ( ~p ) h µ R | log x | dµ ) . Remark that we will finally establish the formula in Theorem 1.1, thus by Propo-sition 3.4 and Proposition 3.5 we havemax (cid:26) , lim inf N →∞ α N (cid:27) = max (cid:26) , lim sup N →∞ α N (cid:27) = max ( , sup µ ∈N ( ~p ) h µ R | log x | dµ ) , and if lim inf N →∞ α N ≥ /
2, then the limit of α N exists and equals tosup µ ∈N ( ~p ) h µ R | log x | dµ . Proof of Proposition α = lim sup N →∞ lim k →∞ sup P ∈ P kN − P p ( i , · · · , i k ) log p ( i , · · · , i k )2 P p ( i , · · · , i k ) log q k ( i , · · · , i k ) . Denote D := lim sup N →∞ lim k →∞ sup P ∈ P kN X p ( i , · · · , i k ) log q k ( i , · · · , i k ) . If D = ∞ , then by Lemma 3.3, we have α ≤ / D < ∞ , which is equivalent tolim sup N →∞ lim k →∞ sup P ∈ P kN Z | log x | dP < ∞ . By (3.5), α = lim sup N →∞ lim k →∞ sup P ∈ P kN h P R | log x | dP . Without loss of generality, we may suppose that there is a sequence of Markovmeasures P N,k ∈ P kN converging to a measure µ ∈ N ( ~p ) in the weak ∗ -topologysuch that lim sup N →∞ lim k →∞ h P N,k R | log x | dP N,k = α. Then by the upper semi-continuity of the entropy function and the weak conver-gence, we have α ≤ sup µ ∈N ( ~p ) h µ R | log x | dµ . (cid:3) Proof of Proposition N which is large enough, and any ǫ >
0, we have E ~p ⊂ ∞ [ ℓ =1 ∞ \ n = ℓ H n ( ǫ, N ) , where H n ( ǫ, N ) := (cid:26) x ∈ [0 , \ Q : (cid:12)(cid:12)(cid:12)(cid:12) τ j ( x, n ) n − p j (cid:12)(cid:12)(cid:12)(cid:12) < ǫ, ≤ j ≤ N (cid:27) . AI-HUA FAN, LING-MIN LIAO, AND JI-HUA MA
For any γ > max { / , α N } and for any integer k ∈ N , we have the γ -Hausdorffmeasure (see [4], for the definition of H γ ) H γ ∞ \ n = ℓ H n ( ǫ, N ) ! ≤ X | τj ( x,n ) n − p j | <ǫ, ≤ j ≤ N | I n ( x ) | γ ( ∀ n ≥ ℓ )= X n ( p j − ǫ )
2, for N large enough ∞ X j = N +1 j + 1) γ < . (3.6)By applying Lemma 2.6, and noticing that τ i ··· i k ( x, ˜ n ) ≤ τ i ··· i k ( x · · · x ˜ n ) + k ,we have | I ( x , · · · , x ˜ n ) | = exp { log | I ( x , · · · , x ˜ n ) |}≤ exp X i ··· i k ∈ Σ kN ( τ i ··· i k ( x · · · x ˜ n ) + k ) log p k ( i , · · · , i k ) q k ( i , · · · , i k ) + 8 + 8˜ n k . Thus X x ··· x ˜ n ∈ ˜ A | I ( x , · · · , x ˜ n ) | γ ≤ X m i ··· ik X x ··· x ˜ n ∈ B exp γ X i ··· i k ∈ Σ kN ( m i ··· i k + k ) log p k q k + 8 γ + 8˜ nγ k , where B := n x · · · x ˜ n ∈ ˜ A : τ i ··· i k ( x · · · x ˜ n ) = m i ··· i k ∀ i · · · i k ∈ Σ kN o . Take h = 1 k X i ··· i k ∈ Σ kN φ (cid:18) m i ··· i k ˜ n − k + 1 (cid:19) (3.7) N THE FREQUENCY OF PARTIAL QUOTIENTS OF REGULAR CONTINUED FRACTIONS9 in Lemma 2.7. We have for any δ > n large enough X x ··· x ˜ n ∈ B exp γ X i ··· i k ∈ Σ kN ( m i ··· i k + k ) log p k q k + 8 γ + 8˜ nγ k ≤ exp ˜ n ( h + δ ) + 2 γ X i ··· i k ∈ Σ kN ( m i ··· i k + k ) log p k q k + 8 γ + 8˜ nγ k . Rewrite the right side of the above inequality asexp { ˜ n ( L ( γ, k, m i ··· i k )) } , where L ( γ, k, m i ··· i k ) := h + 2 γ X i ··· i k ∈ Σ kN m i ··· i k + k ˜ n log p k q k + 8 γ ˜ n + 8 γ k + δ. Since there are at most (˜ n − k + 1) N k possible words of i · · · i k in Σ ˜ nN , we have X x ··· x ˜ n ∈ ˜ A | I ( x , · · · , x ˜ n ) | γ ≤ (˜ n − k + 1) N k exp ( ˜ n sup m i ··· ik L ( γ, k, m i ··· i k ) !) . Notice that by the definition of ˜ A and B , the possible values of m i ··· i k arerestricted to satisfy the condition that the frequency of digit j in x · · · x ˜ n is about p j . Then when ˜ n → ∞ , for i , · · · , i k ∈ Σ kN m i ··· i k ˜ n − k + 1 → p ( i , · · · , i k ) , (3.8)and { p ( i , · · · , i k ) : i , · · · , i k ∈ Σ kN } defines a probability measure P in P kN .Now take δ > k large enough such that8 γ k < δ, (3.9)and γ > − k P p ( i , · · · , i k ) log p ( i , · · · , i k ) + 5 δ − P p ( i , · · · , i k ) log( p k ( i , · · · , i k ) /q k ( i , · · · , i k )) . (3.10)The last inequality comes from the definition of α N , (3.4) and the assumption γ > α N .By (3.8), for sufficiently large ˜ n , we have γ ˜ n < δ and (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) k X i ··· i k ∈ Σ kN φ (cid:18) m i ··· i k ˜ n − k − (cid:19) − k X i ··· i k ∈ Σ kN φ ( p ( i · · · i k )) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) < δ, (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X i ··· i k ∈ Σ kN m i ··· i k + k ˜ n log p k q k − X i ··· i k ∈ Σ kN p ( i · · · i k ) log p k q k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) < δ. (3.11) By (3.7) and (3.11), L ( γ, k, m i ··· i k ) < k X i ··· i k ∈ Σ kN φ ( p ( i · · · i k )) + 2 γ X i ··· i k ∈ Σ kN p ( i · · · i k ) log p k q k + 5 δ. Thus (3.10) implies L ( γ, k, m i ··· i k ) < . Hence finally, by (3.6), we can obtain for any γ > max { / , α N } , H γ ∞ \ n = k H n ( ǫ, N ) ! < ∞ . This implies that dim H ( E ~p ) ≤ max { / , α N } as desired. (cid:3) Lower bound
In this section, we first prove dim H ( E ~p ) ≥ /
2. Then we examine what happensif the condition P ∞ j =1 p j log j < ∞ in Billingsley and Henningsen’s theorem is vio-lated. We will see that if the condition is not satisfied, then dim H ( E ~p ) = 1 / N ( p ) = ∅ .The following is the key lemma for proving that dim H ( E ~p ) ≥ / Lemma 4.1.
For any given sequence of positive integers { c n } n ≥ tending to theinfinity, there exists a point z = ( z , z , . . . ) ∈ E ~p such that z n ≤ c n for all n ≥ .Proof. For any n ≥
1, we construct a probability vector ( p ( n )1 , p ( n )2 , . . . , p ( n ) k , . . . )such that p ( n ) k > ≤ k ≤ c n and P c n k =1 p ( n ) k = 1, and that for any k ≥ n →∞ p ( n ) k = p k . (4.1)Consider a product Bernoulli probability P supported by Q ∞ n =1 { , . . . , c n } . For eachdigit k ≥
1, consider the random variables of x ∈ N N , X i ( x ) = 1 { k } ( x i ), ( i ≥ k , lim n →∞ n n X i =1 { k } ( x i ) − n X i =1 E (1 { k } ( x i )) ! = 0 P − a.s., which implieslim n →∞ n n X i =1 { k } ( x i ) = lim n →∞ n n X i =1 p ( i ) k = p k P − a.s.. (4.2)That is to say, for P almost every point in the space Q ∞ n =1 { , . . . , c n } , the digit k hasthe frequency p k . Considering each point in N N as a continued fraction expansionof a number in [0 , (cid:3) Proof of dim H ( E ~p ) ≥ /
2. Take c n = n in Lemma 4.1, we find a point z ∈ E ~p , suchthat z n = a n ( z ) ≤ n ( ∀ n ≥ . (4.3)For a positive number b >
1, set F z ( b ) := { x ∈ [0 ,
1) : a k ( x ) ∈ ( b k , b k ]; a k ( x ) = a k ( z ) if k is nonsquare } . N THE FREQUENCY OF PARTIAL QUOTIENTS OF REGULAR CONTINUED FRACTIONS11
It is clear that F z ( b ) ⊂ E ~p for all b >
1. We define a measure µ on F z ( b ). For n ≤ m < ( n + 1) , set µ ( I m ( x )) = n Y k =1 b k . (4.4)Denote by B ( x, r ) the ball centered at x with radius r . We will show that for any θ >
0, there exists b >
1, such that for all x ∈ F z ( b ),lim inf r → log µ ( B ( x, r ))log r ≥ − θ. (4.5)In fact, for any positive number r , there exist integers m and n such that | I m +1 ( x ) | < r ≤ | I m ( x ) | and n ≤ m < ( n + 1) . (4.6)By the construction of F z ( b ), a n ( x ) > b n >
1. Let x = [ x , x , · · · ]. By Lemma2.5, B ( x, r ) is covered by the union of three adjacent rank n basic intervals, i.e., B ( x, r ) ⊂ I ( x , x , · · · , x n − ∪ I ( x , x , · · · , x n ) ∪ I ( x , x , · · · , x n + 1) . By the definition of µ , the above three intervals admit the same measure. Henceby (4.6), we havelog µ ( B ( x, r ))log r ≥ log 3 µ ( I ( x , x , · · · , x n ))log | I ( x , x , · · · , x m +1 ) | . (4.7)However, on the one hand, by (4.4) − log µ ( I ( x , x , · · · , x n )) = − log n Y k =1 b k = n X k =1 k log b. (4.8)On the other hand, by (2.2) and Lemma 2.1, we have − log | I ( x , x , · · · , x m +1 ) | ≤ log 2 + m +1 X k =1 x k + 1) . Let us estimate the second term of the sum. First we have m +1 X k =1 x k + 1) ≤ n +1 X k =1 log( x k + 1) + 2 m +1 X k =1 log( z k + 1) . Since x k ≤ b k for all k ≥
1, we deduce m +1 X k =1 log( x k + 1) ≤ n +1 X k =1 log(2 b k + 1) ≤ n +1 X k =1 log(3 b k )= ( n + 1) log 3 + n +1 X k =1 k log b. By (4.3), since z n ≤ n , for all n ≥
1, we know m +1 X k =1 log( z k + 1) ≤ ( n +1) X k =1 log( k + 1) . Thus − log | I ( x , x , · · · , x m +1 ) |≤ log 2 + 2( n + 1) log 3 + 2 n +1 X k =1 k log b + 2 ( n +1) X k =1 log( k + 1) . (4.9)Combining (4.8) and (4.9), for any θ >
0, take b > n →∞ log µ ( I ( x , x , · · · , x n ))log | I ( x , x , · · · , x m +1 ) | ≥ − θ, ∀ x ∈ F z ( b ) . Hence by (4.7), we obtain (4.5).Since θ can be arbitrary small, by Billingsley Theorem ([1]), we havedim H ( E ~p ) ≥ . (cid:3) Thus Theorem 1.1 is already proved under the condition P ∞ j =1 p j log j < ∞ .Now assume P ∞ j =1 p j log j = ∞ . Then for any invariant and ergodic measure µ such that µ ( I ( j )) = p j for all j ≥
1, we have Z | log x | dµ ( x ) ≥ ∞ X j =1 µ ( I ( j )) log j = ∞ X j =1 p j log j = ∞ . (4.10)This implies D = ∞ (see the proof of Proposition 3.5 for the definition of D ).Then by Proposition 3.4, we have dim H ( E ~p ) ≤ . Since we have already proveddim H ( E ~p ) ≥ , we get dim H ( E ~p ) = 12 if ∞ X j =1 p j log j = ∞ . This is in accordance with the formula of Theorem 1.1 under the convention thatsup ∅ = 0, because (4.10) implies N ( ~p ) = ∅ .Finally, we remark that N ( ~p ) = ∅ if and only if P ∞ j =1 p j log j = ∞ . We have seenthe “if” part. For the other part, assume P ∞ j =1 p j log j < ∞ . Then the Bernoullimeasure µ such that µ ( I ( j )) = p j satisfies Z | log x | dµ ( x ) ≤ ∞ X j =1 µ ( I ( j )) log( j + 1) = ∞ X j =1 p j log( j + 1) < ∞ . which implies that N ( ~p ) = ∅ . 5. A remark
As suggested by the referee, we add a remark on a problematic argument ap-pearing in the literature. To obtain an upper bound of the Hausdorff dimension ofa set, one usually applies the Billingsley’s theorem by constructing a finite measure P on the set such that | U | s ≤ P ( U )(see [4], p.67). For the set E ~p , where p j = 0 for some j , the Markov measure P satisfying (3.3) does not match because the cylinders starting with j do not N THE FREQUENCY OF PARTIAL QUOTIENTS OF REGULAR CONTINUED FRACTIONS13 charge the measure and the above inequality is obviously not true. This appearedunnoticed for long (see the remarks of Kifer [7], p. 2012).This problem did exist in the proof of Theorem 2 in [2]. Let us briefly indicatehow to get around the problem in the proof of Theorem 2 of [2] when p j = 0 forsome j ’s. The basic idea is similar to that of Cajar (see [3], p.67) and that of Kifer[7]. Recall that P is a ( k − { x ∈ [0 , ∩ Q c : a n ( x ) ≤ N for all n ≥ } . It is uniquely determined by its values on the k -cylinders, namely, p ( i , . . . , i k ) = P ([ i , . . . , i k ]) , ( i · · · i k ) ∈ { , , · · · , N } k . Let 0 < ǫ < P ǫ be the perturbed ( k − p ǫ ( i , . . . , i k ) = (1 − ǫ ) P ([ i , . . . , i k ]) + ǫN k , ( i · · · i k ) ∈ { , , · · · , N } k . Now, we can apply the Billingsley’s theorem with P ǫ to find an upper bound, andthen get the desired result by letting ǫ → Acknowledgments :
The authors are grateful to the referee for a number ofvaluable comments. This work was partially supported by NSFC10728104 (A. H.Fan) and NSFC10771164 (J. H. Ma).
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Ai-Hua FAN: Department of Mathematics, Wuhan University, Wuhan, 430072, P.R.China & CNRS UMR 6140-LAMFA, Universit´e de Picardie 80039 Amiens, France
E-mail address : [email protected] Ling-Min LIAO: Department of Mathematics, Wuhan University, Wuhan, 430072, P.R.China & CNRS UMR 6140-LAMFA, Universit´e de Picardie 80039 Amiens, France
E-mail address : [email protected] Ji-Hua MA: Department of Mathematics, Wuhan University, Wuhan, 430072, P.R.China
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