Full dimesnsional sets of reals

Full dimesnsional sets of reals whose sums of partial quotients increase in certain speed

Liangang Ma Dept. of Mathematical Sciences, Binzhou University, Huanghe 5th road No. 391, City of Binzhou, Shandong Province, P. R. China
Abstract.

For a real , let be its continued fraction expansion. Let . The Hausdorff dimensions of the level sets

for and a non-decreasing sequence have been studied by E. Cesaratto, B. Vallée, J. Wu, J. Xu, G. Iommi, T. Jordan, L. Liao, M. Rams et al. In this work we carry out a kind of inverse project of their work, that is, we consider the conditions on under which one can expect a -dimensional set . We give certain upper and lower bounds on the increasing speed of when is of Hausdorff dimension 1 and a new class of sequences such that is of full dimension. There is also a discussion of the problem in the irregular case.

Key words and phrases:
Hausdorff dimension; sums of partial quotients; continued fractions
2010 Mathematics Subject Classification:
Primary 11K50; Secondary 37E05, 28A80

1. Introduction

Let be the unit interval. For a real , let

 (1.1) x=[a1(x),a2(x),a3(x),⋯]:=1a1(x)+1a2(x)+1a3(x)+⋱

be its continued fraction expansion. Let

be the Gauss map, with the two symbols and being the fractional and integral part of the number. is conjugated to a shift map on a countable alphabet. Let

be the sum of the first partial quotients, . We focus on the limit behaviors of in this work. According to A. Ya. Khinchin [Khi],

almost everywhere with respect to Lebesgue measure. In 1988, W. Philipp [Phi, Theorem 1] strengthened Khinchin’s result by showing that, for a sequence of positive numbers such that is non-decreasing,

or a. e.

according to whether or . His proof relies on the theory of mixing random vertors or triangular arrays. As to subsets of the residual set, which are all of measure 0, it turns out that the Hausdorff dimension is a useful tool to distinguish their sizes. From the point view of dynamical systems , E. Cesaratto and B. Vallée [CV], G. Iommi and T. Jordan [IJ] got interesting results on Hausdorff dimension of the sets

or

for , as applications of their more comprehensive results in their more general contexts (the case is also computed in [IJ]).

The level sets

 (1.2) Eφ(n),α:={x∈(0,1):limn→∞sn(x)φ(n)=α}

for and non-decreasing are considered by J. Xu [Xu], J. Wu and J. Xu [WX1], as well as L. Liao and M. Rams [LR1]. For a set , let be its Hausdorff dimension. They proved various results on with different increasing speed . For example, in [WX1], Wu and Xu showed that

for any . They also gave more sequences such that in [WX1, 4] with the restriction that

 (1.3) limsupn→∞loglogφ(n)logn<12

In the case , Liao and Rams [LR1, Theorem 1.2] proved that a full dimensional set is still possible for some sequences . In this paper we continue the search for non-decreasing sequences such that , as an attempt to exhausting sequences with this property. We first show that

1.1 Theorem.

For a non-decreasing sequence and a non-negative real , if , then

and .

Then we continue to point out that

1.2 Theorem.

For every , there exists a non-decreasing sequence , such that

and .

Remark.

Examples of with have been given in [WX1] and [LR1] according to their results mentioned before. Our examples (together with [LR1, Theorem 1.2]) with shows how big effect a mild change on the growth rate of can have on the dimension of the level sets . One is recommended to compare these results with [WX1, 4] and [LR1, Theorem 1.1, 1.2].

At last, for the irregular case (see Section 5), we show that

1.3 Theorem.

For any small , there exists a non-decreasing sequence with

and

,

such that

.

As proofs of these results for any are of completely same processes (case is usually trivial by some known results), we only deal with dimension of the set instead of in the following.

2. Some notations and established results

By a rank- basic interval we mean

for a fixed sequence of positive integers . Its length can be explicitly expressed as a function of . If one sets and for , then

.

Especially, we use the following estimation,

.

According to I. G. Good [Goo, P209], a covering set of intervals whose elements are all basic intervals is called a fundamental covering system. We always use fundamental covering systems throughout the work. It is enough to restrict the coverings to be fundamental ones on discussing dimensions of sets in continued fractions. The following result says that adding or neglecting finitely many partial quotients does not affect the dimension of the set. <ltx:theorem xml:id='#id' inlist='thm theorem:Good'scorollary' class='ltx_theorem_Goodscorollary'>#tags<ltx:title font='#titlefont' _force_font='true'>#title</ltx:title>#body</ltx:theorem>

For a sequence of positive numbers , let

in which reads infinitely often. Good [Goo] have ever gave bounds on for some , T. Łuczak [Luc, Theorem] (see also [FWLT]) showed for any . In 2008, B. Wang and J. Wu [WW] determind precise values of for any , which greatly strengthens Good and Łuczak’s results. They proved that <ltx:theorem xml:id='#id' inlist='thm theorem:WW'sTheorem' class='ltx_theorem_WWsTheorem'>#tags<ltx:title font='#titlefont' _force_font='true'>#title</ltx:title>#body</ltx:theorem> If we define as the pressure function, then in the theorem. One is recommended to [MU] and [Wal] for more general pressure functions.

Now we introduce some notations and results by E. Cesaratto and B. Vallée [CV] as well as G. Iommi and T. Jordan [IJ]. We only state their results in some simple cases, which will be enough for our uses. Their original results are in far more general contexts. Denote by , is the Euler constant. Then according to [CV, Theorem 2], <ltx:theorem xml:id='#id' inlist='thm theorem:CV'stheorem' class='ltx_theorem_CVstheorem'>#tags<ltx:title font='#titlefont' _force_font='true'>#title</ltx:title>#body</ltx:theorem> While their result is focused essentially on the exponential convergent rate of to as , we only use the fact for any . By [IJ, Corollary 6.6, Proposition 6.7] we have <ltx:theorem xml:id='#id' inlist='thm theorem:IJ'sproposition' class='ltx_theorem_IJsproposition'>#tags<ltx:title font='#titlefont' _force_font='true'>#title</ltx:title>#body</ltx:theorem> This means that for any there exists an unique such that . The Proposition will be exploited in our Proposition 5.3. At last we recover a result of A. Fan, L. Liao, B. Wang and J. Wu [FLWW1, Lemma 3.2] as following. <ltx:theorem xml:id='#id' inlist='thm theorem:FLWW'slemma' class='ltx_theorem_FLWWslemma'>#tags<ltx:title font='#titlefont' _force_font='true'>#title</ltx:title>#body</ltx:theorem> The lemma is generalized to the following form by Liao and Rams [LR2, Lemma 2.3]. <ltx:theorem xml:id='#id' inlist='thm theorem:LR'sLemma' class='ltx_theorem_LRsLemma'>#tags<ltx:title font='#titlefont' _force_font='true'>#title</ltx:title>#body</ltx:theorem> They are very useful in dealing with sets with dimensions . For various applications of them, see [FLWW1] [JR] [LR1] and [LR2].

3. Proof of Theorem 1.1

In this section we prove the two necessary conditions on the growth rate of for the set to be of dimension 1. We first show the requirement on the lower growth rate.

3.1 Lemma.

For a non-decreasing sequence , if , then

.

Proof.

We show this by reduction to absurdity. Suppose that there exists a non-decreasing sequence , such that and . Then for this ,

according to our notations before. By CV’s Theorem, , so . This contradicts the assumption , so the conclusion follows. ∎

Remark.

As far as the author knows, all the examples of sequences wtih satisfies . However, among the slightly lower growing sequences with and , we are not sure whether there exists one such that , see the discussions in Section 5.

Now we show the bound on the upper growth rate of for sequences with .

3.2 Theorem.

For a non-decreasing sequence , if , then

.

Proof.

We again show this by reduction to absurdity. Suppose that there exists a non-decreasing sequence , such that and . Then we can find some and small , such that for some and all ,

.

So we have . By choosing large enough, we can guarantee that for all ,

.

Now consider the set for this . If , then for large enough, without loss of generality (by Good’s Corollary), suppose for all ,

.

Then for all ,

.

Now split into two sets and ,

,

.

Clearly we have

 (3.1) Eφ(n)=E1∪E2.

By WW’s Theorem,

 (3.2) dimHE1<1.

For , note that

 (3.3) E2=∪∞k=1{x∈Eφ(n):ai(x)≤Bi for % all i≥k}

We claim that

for any . Considering (3.3) This will force

 (3.4) dimHE2≤1/2

We only show the claim in the case , proofs for the general cases are similar. When , let

.

We have the following decomposition:

This is because if there is an infinite sequence , such that , then for any fixed and , the inequality

can not hold for all , considering the two restrictions for all and for all . Now by FLWW’s Lemma,

.

So the claim is true, which successively justifies (3.4).

Finally, combining (3.1) (3.2) and (3.4), we get

under the assumptions on the sequence . This contradicts the assumption that , so the theorem follows.

Remark.

In an earlier version of the paper, the author showed that a non-decreasing sequence with must satisfy

in which is the Riemann zeta function, based on [Goo, Theorem 5]. Shortly after Prof. B. Wang told me that the bound can be decreased from to by WW’s Theorem. The proof now is modified according to his comments. Prof. Wang also gave an easier proof of the theorem, however, we feel the old one still contains something interesting and the idea will be retrieved briefly in Corollary 6.1, so we decide only to modify it to some extent. Here is Prof. Wang’s proof of Theorem 3.2: suppose there exists a non-decreasing sequence with and , then for some small and , we have

for large enough. If , then for large enough, . So there exists , such that . This implies

.

Then by WW’s Theorem, , which contradicts the assumptions.

It is easy to see that, from WW’s Theorem, we can show

3.3 Corollary.

For a non-decreasing sequence and , if , then

.

3.4 Corollary.

For a non-decreasing sequence and , if , then

.

There are more discussions on the techniques used in the proof of Theorem 3.2 in Section 6.

Proof of Theorem 1.1:

Proof.

This is an instant corollary of Lemma 3.1 and Theorem 3.2. ∎

4. Proof of Theorem 1.2

In this section we give an explicit non-decreasing sequence satisfying the properties in Theorem 1.2, based on Wu and Xu’s work [WX1]. For , take a sequence . For , let

Without special declaration we always mean this sequence by the notation in this section. We will show that

4.1 Proposition.

For large enough and the sequence defined above, we have

and .

It is easy to check that . In the following we will demonstrate that . The proof follows Wu and Xu’s method [WX1, 3], we only give proofs where we feel necessary. We first give some notations following Wu and Xu. For an , let

.

We will omit the integer notation in the following for simplicity, as results will not be affected. It is easy to show that . For any , let

.

Obviously we have

.

Let , . For any , let be the finite sequence by deleting the terms in . Let be the finite continued fraction of . Now we show that

4.2 Lemma.

For any , there exists , such that for any and , we have

.

Proof.

Assume for some . First, by [Wu, Lemma 2.1], we have

 (4.1) |In(a1,⋯,an)|≥12q2n(a1,⋯,an)≥12(qn−t(n)¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯(a1,⋯,an)(enβk−1−enβk−2+1)t(n))2.

Moreover,

 (4.2) 2(enβk−1−enβk−2+1)2t(n)≤2(enβk−1)2t(n)≤2e2nβ(1/nN)≤\textcircled12(n−t(n)−1)ϵ≤\textcircled2q2ϵn−t(n)¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯(a1,⋯,an).

The inequality 1⃝ holds as long as is large enough, while 2⃝ is due to for any finite -continued fraction. Now combining (4.1) and (4.2) we have

.

Remark.

From the proof one can see that the inequlity 1⃝ can hold only for no matter how small is (recall describes density of the terms in ). This is the obstacle to generalize the method to cases .

Now for two points and , there is one and only one integer , such that

with

.

For the distance between and , similar to [WX1, Lemma 3.4], we can show

.

Now we are in a position to prove Proposition 4.1.

Proof of Proposition 4.1:

Proof.

This is of similar process as [WX1, Proof of Theorem 1.4], with only a change of some constants. ∎

Theorem 1.2 follows directly from Proposition 4.1.

5. The irregular case

While cases of sums of partial quotients grow linearly () were considered in [CV] [IJ], Wu and Xu [WX1], Liao and Rams [LR1] have dealt with the dimentional problem under the regular condition on the growth rate. It is termed the super-linear case in [WX1]. The restriction does guarantee some convenience and general results (the condition is also assumed in [FLWW2] and [LR2]) on studying . However, there seems little known in the irregular case that satisfies

 (5.1) liminfn→∞φ(n)n<∞

and

 (5.2) limsupn→∞φ(n)n=∞.

simutaneously. In this section we present some results in this case, by establishing some links between results on with various growth rates . However, we are not the first to deal with some irregular growth rate considering [DV, Corollary 3] in continued fractions. Note that our sequences are even more “irregular” than ones in [DV] as can not be non-decreasing for satisfying 5.1 and 5.2.

For two -sequences and in differs only at the subscripts , we mean

and

for , . Compare the denominators of the convergents, we have

Proof.

Let and for . We only show

 (5.3) qn(→σn)qn(→τn)<(σn1+1)(σn2+1)⋯(σnt(n)+1)τn1τn2⋯τnt(n),

the other inequality can be shown similarly. First, for as for . When ,

.

When ,

,

so

.

Inductively one can show

for any . When ,

.

By similar inductive steps as before we can show

for any . The inequality 5.3 holds by inductive processes on for . ∎

Now we aim to compare lengths of the two rank- basic intervals and .

Comparison Lemma.

Suppose that for large enough

 (5.4) max{σn1+1,σn2+1,⋯,σnt(n)+1,τn1+1