A remark on the extreme value theory for continued fractions

# A remark on the extreme value theory for continued fractions

Lulu Fang School of Mathematics, South China University of Technology, Guangzhou 510640, P.R. China  and  Kunkun Song School of Mathematics and Statistics, Wuhan University, Wuhan, 430072, P.R. China
###### Abstract.

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

 Eϕ={x∈(0,1):limn→∞Tn(x)ϕ(n)=1},

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

## 1. Introduction

Every irrational number in the unit interval has a unique continued fraction expansion of the form

 x=1a1(x)+1a2(x)+⋱+1an(x)+⋱, (1.1)

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 [11].

Let be an irrational number. For any , we define

 Tn(x)=max{ak(x):1≤k≤n},

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 [12]). Galambos [5] 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

 μ(A)=1log2∫A11+xdx

for any Borel set . Later, he also gave an iterated logarithm type theorem for in [6], that is, for -almost all ,

 limsupn→∞logTn(x)−lognloglogn=1    and%     liminfn→∞logTn(x)−lognloglogn=0.

As a consequence, we know that

 limn→∞logTn(x)logn=1

holds for -almost all . Furthermore, Philipp [16] solved a conjecture of Erdős for and obtained its order of magnitude. In fact,

 liminfn→∞Tn(x)loglognn=1log2.

holds for -almost all . Later, Okano [15] 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,

 limsupn→∞Tn(x)bn=0        or        limsupn→∞Tn(x)bn=+∞

holds for -almost all according to converges or diverges. In 1935, Khintchine [10] proved that converges in measure to the constant and Philipp [17] 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 [3] showed that the maximum should be responsible for the failure of the strong law of large numbers, i.e.,

 limn→∞Sn(x)−Tn(x)nlogn=1log2

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 [1], Bazarova et al. [2], Iosifescu and Kraaikamp [7] 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

 Eϕ={x∈(0,1):limn→∞Tn(x)ϕ(n)=1},

where is a positive function defined on with as . Wu and Xu [19] 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 ([19]).

Assume that is a positive function defined on satisfying as and

 limn→∞logϕ(n)logn<∞.

Then

Liao and Rams [13] 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 ([13]).

For any ,

 dimH{x∈(0,1):limn→∞Tn(x)enα=β}={1,α∈(0,1/2);1/2,α∈(1/2,+∞).

However, they don’t know what will happen at the critical point . Recently, Ma [14] 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 [19], Liao and Rams [13], and Ma [14] in this topic.

###### Theorem 1.3.

Let be real numbers. Then for any ,

 dimH{x∈(0,1):limn→∞Tn(x)cbnα=β}=⎧⎨⎩1/2,α∈(0,1);1/(b+1),α=1;0,α∈(1,+∞).

For more results about the Hausdorff dimensions of fractal sets related to , see Zhang [21], and Zhang and Lü [22]. The following figure is an illustration of the Haudorff dimension of for different .

## 2. Preliminaries

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

 pn(x)qn(x):=[a1(x),a2(x),⋯,an(x)]

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:

 pn(x)=an(x)pn−1(x)+pn−2(x)   and   qn(x)=an(x)qn−1(x)+qn−2(x), (2.1)

which implies that

 a1(x)a2(x)⋯an(x)≤qn(x)≤(a1(x)+1)(a2(x)+1)⋯(an(x)+1). (2.2)
###### Definition 2.1.

For any and , we call

 I(a1,⋯,an):={x∈(0,1):a1(x)=a1,⋯,an(x)=an}

the -th order cylinder of continued fraction expansion.

In other words, is the set of points beginning with in their continued fraction expansions.

###### Proposition 2.2.

Let and . Then is an interval with two endpoints

 pnqn     and     pn+pn−1qn+qn−1.

More precisely, is the left endpoint if is even; otherwise it is the right endpoint. Moreover, the length of satisfies

 |I(a1,⋯,an)|=1qn(qn+qn−1), (2.3)

where and satisfy the recursive formula (2.1).

## 3. Proof of Theorem 1.3

In this section, we will give the proof of Theorem 1.3 which is inspired by Xu [20]. More precisely, we will determine the Hausdorff dimension of the fractal set

 E(b,c,α,β)={x∈(0,1):limn→∞Tn(x)cbnα=β}

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

 E∗(b,c,α):={x∈(0,1):an(x)≥cbnαi.o.},

where denotes infinitely often. The following result is a special case of Theorem 4.2 in Wang and Wu [18].

###### Lemma 3.1.

([18, Theorem 4.2]) Let be a real number. Then

 dimHE∗(b,c,α)=⎧⎨⎩1/2,α∈(0,1);1/(b+1),α=1;0,α∈(1,+∞).

Now we show that is a subset of with .

###### Lemma 3.2.

Let and . Then for any , we have .

###### Proof.

Since for any , we know that for sufficiently large . Choose enough small such that .

For any , we know that

 limn→∞Tn(x)cbnα=β.

Note that

 (β−δ)cb(n+1)α−(β+δ)cbnαcd(n+1)α =cb(n+1)αcd(n+1)α⋅(β−δ−((β+δ)⋅bnαb(n+1)α)) ≥cb(n+1)α−d(n+1)α⋅(β−δ−(β+δ)⋅1c)→∞,

then there exists (depending on ) such that for any , we have

 (β−δ)cbnα≤Tn(x)≤(β+δ)cbnα and  (β−δ)cb(n+1)α−(β+δ)cbnα≥cd(n+1)α.

Thus, we actually deduce that

 an+1(x)≥Tn+1(x)−Tn(x)≥(β−δ)cb(n+1)α−(β+δ)cbnα≥cd(n+1)α

holds for any . So we have . ∎

It follows from Lemma 3.1 that for and for . When , by Lemmas 3.1 and 3.2, we know that

 dimHE(b,c,α,β)≤dimHF(d,c,α)≤1/(d+1)

for any . So by letting . Therefore, we eventually obtain that

 dimHE(b,c,α,β)≤⎧⎨⎩1/2,α∈(0,1);1/(b+1),α=1;0,α∈(1,+∞).

### 3.2. Lower bound

For any , we define

 f(n)=(β−1n)cbnα    and     g(n)=(β+1n)cbnα.

Then and as . So we can choose sufficiently large such that

 f(n)≥2, g(n) is non-decreasing and  cbnαn≥2

for all . Let

 FN(b,c,α,β)={x∈(0,1):f(n)≤an(x)≤g(n) % for all n≥N}.

The following lemma states that is a subset of .

.

###### Proof.

For any , we have

 f(n)≤an(x)≤g(n) (3.1)

for all . Note that is non-decreasing for all , we know that

 Tn(x)≤max{TN−1(x),g(n)}≤TN−1(x)+g(n).

Combing this with (3.1), we deduce that

 f(n)≤an(x)≤Tn(x)≤TN−1(x)+g(n)

for any . Therefore,

 limn→∞Tn(x)cbnα=β

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]).

###### Lemma 3.4.

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

 dimHE≥liminfn→∞log(m1m2⋯mn−1)−log(mnεn).
###### Proof.

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

 μ(U) ≤min{mn,2|U|/εn}⋅(m1⋯mn)−1≤m1−sn⋅(2|U|/εn)s⋅(m1⋯mn)−1

for any and hence

 μ(U)|U|s≤2s(m1⋯mn−1)msnεsn.

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. ∎

###### Lemma 3.5.
 dimHFN(b,c,α,β)≥⎧⎨⎩1/2,α∈(0,1);1/(b+1),α=1;0,α∈(1,+∞).
###### Proof.

For any , we define

 Dn={(σ1,⋯,σn)∈Nn: 1≤σk≤ cbkα/k+1 for all 1≤k

and

 D=⋃n≥0Dn

with the convention . For any and , we denote

 J(σ1,⋯,σn)=⋃σn+1clI(σ1,⋯,σn,σn+1)

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.

Let

 E0=[0,1],    En=⋃(σ1,⋯,σn)∈DnJ(σ1,⋯,σn)

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

 Mn=⌊cbnαn⌋+1,

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

 |I(τ1,⋯,τn,1)|  or  |I(σ1,⋯,σn,1)|,

where denotes the length of a interval. In view of (2.2) and (2.3), we deduce that

 |I(τ1,⋯,τn,1)|≥12q2n+1 ≥18⋅(n∏k=1(τk+1))−2 =122n+3⋅(N−1∏k=1(cbkαk+1)n∏k=N(β+1k)cbkα)−2 :=εn.

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

 dimHE ≥liminfn→∞log(m1m2⋯mn)−log(mn+1εn+1) ≥liminfn→∞b1α+b2α+⋯+bnα2(b1α+⋯+b(n+1)α)−b(n+1)α =liminfn→∞b1α+⋯+bnα2(b1α+⋯+bnα)+b(n+1)α.

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.

Combing Lemmas 3.3 and 3.5, we get

 dimHE(b,c,α,β)≥dimHFN(b,c,α,β)≥⎧⎨⎩1/2,α∈(0,1);1/(b+1),α=1;0,α∈(1,+∞).

The results on the upper and lower bounds imply the proof of Theorem 1.3.

###### Remark 3.6.

In fact, from the proof of the lower bound, we can obtain that

 dimHEϕ≥liminfn→∞logϕ(1)+⋯+logϕ(n)2(logϕ(1)+⋯+logϕ(n))+logϕ(n+1).

Therefore, we always have when tends to infinity with single exponential rates since

 limsupn→∞(n+1)α1α+2α+⋯+nα=0

for any . And we also get when and if since

 limsupn→∞b(n+1)αb1α+b2α+⋯+bnα=0

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 .

## References

• [1] D. Barbolosi, Sur l’ordre de grandeur des quotients partiels du développement en fractions continues régulières (French), Monatsh. Math. 128 (1999), no. 3, 189–200.
• [2] A. Bazarova, I. Berkes and L. Horváth, On the extremal theory of continued fractions, J. Theoret. Probab. 29 (2016), no. 1, 248–266.
• [3] H. Diamond and J. Vaaler, Estimates for partial sums of continued fraction partial quotients, Pacific J. Math. 122 (1986), no. 1, 73–82.
• [4] K. Falconer, Fractal Geometry: Mathematical Foundations and Applications, John Wiley & Sons, Ltd., Chichester, 1990.
• [5] J. Galambos, The distribution of the largest coefficient in continued fraction expansions, Quart. J. Math. Oxford Ser. (2) 23 (1972), 147–151.
• [6] J. Galambos, An iterated logarithm type theorem for the largest coefficient in continued fractions, Acta Arith. 25 (1973/74), 359–364.
• [7] M. Iosifescu and C. Kraaikamp, Metrical Theory of Continued Fractions. Mathematics and Its Applications, Kluwer Academic Publishers, Dordrecht, 2002.
• [8] M. Kesseböhmer and M. Slassi, Large deviation asymptotics for continued fraction expansions, Stoch. Dyn. 8 (2008), no. 1, 103–113.
• [9] M. Kesseböhmer and M. Slassi, A distributional limit law for the continued fraction digit sum, Math. Nachr. 281 (2008), no. 9, 1294–1306.
• [10] A. Khintchine, Metrische Kettenbruchprobleme, Compositio Math. 1 (1935), 361–382.
• [11] A. Khintchine, Continued Fractions, The University of Chicago Press, Chicago, 1964.
• [12] M. Leadbetter, G. Lindgren and H. Rootzén, Extremes and related properties of random sequences and processes, Springer Series in Statistics. Springer-Verlag, New York-Berlin, 1983.
• [13] L.-M. Liao and M. Rams, Subexponentially increasing sums of partial quotients in continued fraction expansions, Math. Proc. Cambridge Philos. Soc. 160 (2016), no. 3, 401–412.
• [14] L.-G. Ma, A remark on Liao and Rams’ result on distribution of the leading partial quotient with growing speed in continued fractions, arXiv:1606.03344.
• [15] T. Okano, Explicit continued fractions with expected partial quotient growth, Proc. Amer. Math. Soc. 130 (2002), no. 6, 1603–1605.
• [16] W. Philipp, A conjecture of Erdős on continued fractions, Acta Arith. 28 (1975/76), no. 4, 379–386.
• [17] W. Philipp, Limit theorems for sums of partial quotients of continued fractions, Monatsh. Math., 105 (1988), no. 3, 195–206.
• [18] B.-W. Wang and J. Wu, Hausdorff dimension of certain sets arising in continued fraction expansions, Adv. Math. 218 (2008), no. 5, 1319–1339.
• [19] J. Wu and J. Xu, The distribution of the largest digit in continued fraction expansions, Math. Proc. Cambridge Philos. Soc. 146 (2009), no. 1, 207–212.
• [20] J. Xu, On sums of partial quotients in continued fraction expansions, Nonlinearity 21 (2008), no. 9, 2113–2120.
• [21] Z.-L. Zhang, The relative growth rate of the largest digit in continued fraction expansions, Lith. Math. J. 56 (2016), no. 1, 133–141.
• [22] Z.-L. Zhang and M.-Y. Lü, The relative growth rate of the largest partial quotient to the sum of partial quotients in continued fraction expansions, J. Number Theory 163 (2016), 482–492.
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