A remark on the extreme value theory for continued fractions
Let be a irrational number in the unit interval and denote by its continued fraction expansion . For any , write . We are interested in the Hausdorff dimension of the fractal set
where is a positive function defined on with as . Some partial results have been obtained by Wu and Xu, Liao and Rams, and Ma. In the present paper, we further study this topic when tends to infinity with a doubly exponential rate as goes to infinity.
Key words and phrases:Continued fractions, Extreme value theory, Doubly exponential rate, Hausdorff dimension
2010 Mathematics Subject Classification:Primary 11K50, 28A80; Secondary 60G70
Every irrational number in the unit interval has a unique continued fraction expansion of the form
where are positive integers and are called the partial quotients of the continued fraction expansion of . Sometimes we write the representation (1.1) as . For more details about continued fractions, we refer the reader to a monograph of Khintchine .
Let be an irrational number. For any , we define
i.e., the largest one in the block of the first partial quotients of the continued fraction expansion of . Extreme value theory in probability theory is concerned with the limit distribution laws for the maximum of a sequence of random variables (see ). Galambos  first considered the extreme value theory for continued fractions and obtained that
for any , where is equivalent to the Lebesgue measure, namely Gauss measure given by
for any Borel set . Later, he also gave an iterated logarithm type theorem for in , that is, for -almost all ,
As a consequence, we know that
holds for -almost all . Furthermore, Philipp  solved a conjecture of Erdős for and obtained its order of magnitude. In fact,
holds for -almost all . Later, Okano  constructed some explicit real numbers that satisfied this liminf. However, a natural question arises: what is the exact growth rate of or whether there exists a normalizing sequence such that converges to a positive and finite constant for -almost all . Unfortunately, there is a negative answer for this question. That is to say, there is no such a normalizing non-decreasing sequence so that converges to a positive and finite constant for -almost all . More precisely,
holds for -almost all according to converges or diverges. In 1935, Khintchine  proved that converges in measure to the constant and Philipp  remarked that this result cannot hold for -almost all , where . That is to say, the strong law of large numbers for fails. However, Diamond and Vaaler  showed that the maximum should be responsible for the failure of the strong law of large numbers, i.e.,
for -almost all . These results indicate that the maximum play an important role in metric theory of continued fractions. For more metric results on extreme value theory for continued fractions, we refer the reader to Barbolosi , Bazarova et al. , Iosifescu and Kraaikamp  and Kesseböhmer and Slassi [8, 9].
The following will study the maximum from the viewpoint of the fractal dimension. More precisely, we are interested in the Hausdorff dimension of the fractal set
where is a positive function defined on with as . Wu and Xu  have obtained some partial results on this topic and they pointed out that has full Hausdorff dimension if tends to infinity with a polynomial rate as goes to infinity.
Theorem 1.1 ().
Assume that is a positive function defined on satisfying as and
Liao and Rams  considered the Hausdorff dimension of when tends to infinity with a single exponential rate. Also, they showed that there is a jump of the Hausdorff dimensions from 1 to 1/2 on the class at .
Theorem 1.2 ().
For any ,
However, they don’t know what will happen at the critical point . Recently, Ma  solved this left unknown problem and proved that its Hausdorff dimension is 1/2 when .
In the present paper, we will further investigate the Hausdorff dimension of when tends to infinity with a doubly exponential rate. Moreover, we will see that the Hausdorff dimension of will decay to zero if the speed of is growing faster and faster, which can be treated as a supplement to Wu and Xu , Liao and Rams , and Ma  in this topic.
Let be real numbers. Then for any ,
This section is devoted to recalling some definitions and basic properties of continued fractions.
Let be a irrational number and its continued fraction expansion . For any , we denote by
the -th convergent of the continued fraction expansion of , where and are relatively prime. With the conventions , , , , the quantities and satisfy the following recursive formula:
which implies that
For any and , we call
the -th order cylinder of continued fraction expansion.
In other words, is the set of points beginning with in their continued fraction expansions.
Let and . Then is an interval with two endpoints
More precisely, is the left endpoint if is even; otherwise it is the right endpoint. Moreover, the length of satisfies
where and satisfy the recursive formula (2.1).
3. Proof of Theorem 1.3
for and . The proof is divided into two parts: upper bound and lower bound for .
3.1. Upper bound
Let be real numbers. For any , we define
where denotes infinitely often. The following result is a special case of Theorem 4.2 in Wang and Wu .
([18, Theorem 4.2]) Let be a real number. Then
Now we show that is a subset of with .
Let and . Then for any , we have .
Since for any , we know that for sufficiently large . Choose enough small such that .
For any , we know that
then there exists (depending on ) such that for any , we have
Thus, we actually deduce that
holds for any . So we have . ∎
3.2. Lower bound
For any , we define
Then and as . So we can choose sufficiently large such that
for all . Let
The following lemma states that is a subset of .
For any , we have
for all . Note that is non-decreasing for all , we know that
Combing this with (3.1), we deduce that
for any . Therefore,
and hence . That is to say, . ∎
Next we estimate the lower bound for the Hausdorff dimenson of . To do this, we need the following lemma, which provides a method to obtain a lower bound Hausdorff dimension of a fractal set (see [4, Example 4.6]).
Let , where is a decreasing sequence of subsets in and is a union of a finite number of disjoint closed intervals (called -th level intervals) such that each interval in contains at least intervals of which are separated by gaps of lengths at least . If and , then
Suppose the liminf is positive, otherwise the result is obvious. We may assume that each interval in contains exactly intervals of since we can remove some excess intervals to get smaller sets and . Thus the gaps of lengths between different intervals in are not changed and hence we just need to deal with these new smaller sets. Now we define a mass distribution on by assigning a mass of to each of -th level intervals in . Next we will check the conditions of the classical Mass distribution principle.
Let be an interval of length satisfying . Then there exists a integer such that . On the one hand, can intersect at most one -th level interval since the gap of length between different intervals in is at least and hence can intersect at most -th level intervals; on the other hand, since and the gap of length between different intervals in is at least , we know can intersect at most -th level intervals. Note that each -th level interval has mass , so
for any and hence
Let . Then for sufficiently large . Thus , where is an absolute constant. By the classical Mass distribution principle (see [4, Chapter 4]), we have . This completes the proof. ∎
For any , we define
with the convention . For any and , we denote
and call it the -th level interval, where the union is taken over all such that , cl denotes the closure of a set and is the -th cylinder for continued fractions.
for any and . Then is a Cantor-like subset of . It follows from the construction of that each element in contains some number of the -th level intervals in . We denote such a number by . If , by the definition of , we have
where denotes the greatest integer not exceeding . When , we get
So we obtain .
Next we estimate the gaps between the same order level intervals. For any and two distinct level intervals and of , we assume that locates in the left of without loss of generality. By Proposition 2.2, we know the level intervals and are separated by the -th cylinder or according to is even or odd. In fact, if is odd, note that is a union of a finite number of the closure of -th order cylinders like with and these cylinders run from right to left, and so is , therefore and are separated by in this case. When is even, they are separated by . Thus the gap is at least
Similarly, we can also obtain . It is easy to check that for sufficiently large and as . These imply that the gaps between any two -th level intervals are at least . By Lemma 3.4, we have
When , then as and hence . If , we know as and hence . When , we have as and hence . Since is a subset of , we complete the proof.
The results on the upper and lower bounds imply the proof of Theorem 1.3.
In fact, from the proof of the lower bound, we can obtain that
Therefore, we always have when tends to infinity with single exponential rates since
for any . And we also get when and if since
for and it is when . Moreover, it also indicates that the Hausdorff dimension of is just related to the second base (i.e., ) in the doubly exponential rate when .
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