Dimension Estimates for Certain Sets of Infinite Complex Continued Fractions

Abstract

We prove upper and lower estimates on the Hausdorff dimension of sets of infinite complex continued fractions with finitely many prescribed Gaussian integers. Particulary we will conclude that the dimension of theses sets is not zero or two and there are such sets with dimension greater than one and smaller than one.

1. Introduction

Continued fractions were studied in a number of theories since the work of Wallis in the 17th century; see [1]. The first dimensional theoretical perspective on infinite real continued fractions can be found in the work of Jarnik [2], who introduced upper and lower estimates on the Hausdorff dimension of sets of continued fractions with bounded digits. The problem of calculating the dimension of these sets has been addressed by several authors [3–7]. In a resent work Jenkinson and Pollicott provide a fast algorithm to approximate this dimension [8].

Dimension theoretical aspects of infinite complex continued fractions were studied by Mauldin et al. [9, 10]. They proved that the set of complex continued fractions with arbitrary Gaussian integers from ℕ+ℤ𝑖 has Hausdorff dimension greater than one and smaller than two.

We consider here infinite complex continued fractions and ask for the Hausdorff dimension of the set of continued fractions with digits coming from a finite set 𝐴⊂ℕ[𝑖]. Using the Moran formula from the theory of iterated function systems [11] we are able to give upper and lower estimates on the Hausdorff dimension of these sets; see Theorem 1. We will show that the dimension of the sets is not zero or two and there are such sets with dimension greater than one and smaller than one; see Corollaries 2 and 3. In addition we provide explicit estimates in selected examples.

2. Notations, Results, and Examples

Given a sequence 𝑧_𝑛∈ℂ for 𝑛∈ℕ₀ of Gaussian integers we define the infinite complex continued fraction by 𝑧₀;𝑧₁,𝑧₂,…∶=𝑧₀+11+𝑧₁/1+𝑧₂⋯∈ℂ. (1)

It is well known that every complex number can be represented as an infinite continued fraction of Gaussian integers using the Hurwitz algorithm [12]. Now fix a finite set 𝐴=𝑎_𝑗+𝑏_𝑗𝑖∣𝑎_𝑗,𝑏_𝑗∈ℕ,𝑗=1,…,𝑚⊂ℕ[𝑖]. (2)

We consider the set of all infinite continued fractions having fractional entries coming from 𝐴: ℭ(𝐴)∶=0;𝑧₁,𝑧₂,…∣𝑧_𝑛∈𝐴,𝑛∈ℕ⊂ℂ. (3)

Obviously the set ℭ(𝐴) is uncountable, and it is a null set with respect to the two-dimension Lebesgue measure (this is immediate from Corollary 2). Thus we are interested in the Hausdorff dimension of this set. Recall [13, 14] that the 𝑑-dimensional Hausdorff measure of a set ℭ⊆ℂ is 𝐻^𝑑(ℭ)=lim_𝜖→0inf^∞_𝑖=1diameter𝐶_𝑖^𝑑∣ℭ⊆^∞_𝑖=1𝐶_𝑖,diameter𝐶_𝑖<𝜖. (4)

The Hausdorff dimension of ℭ is given by dim_𝐻ℭ=sup𝑑∣𝐻^𝑑(ℭ)=∞=inf𝑑∣𝐻^𝑑(ℭ)=0. (5)

Now we are able to state our main result on dim_𝐻ℭ(𝐴).

Theorem 1. For a finite set 𝐴⊂ℕ[𝑖] let 𝐷,𝑑∈ℝ⁺ be the unique real numbers fulfilling _{𝑎+𝑏𝑖∈𝐴}1𝑎²+𝑏^2𝐷=1,_{𝑎+𝑏𝑖∈𝐴}1𝑎²+𝑏²+1+√2max{𝑎,𝑏}+1^𝑑=1. (6)
We have 𝑑≤dim_𝐻ℭ(𝐴)≤𝐷. (7)

Unsing an additionla argument this theorem has the following corollary.

Corollary 2. For all finite sets 𝐴⊂ℕ[𝑖] one has dim_𝐻ℭ(𝐴)<2; on the other hand dim_𝐻ℭ(𝐴)>0 if 𝐴 has more than one element.

Proof. Consider _{𝑎+𝑏𝑖∈𝐴}1𝑎²+𝑏²²≤^∞∞_𝑎=1𝑏=11𝑎²+𝑏²²≤14+^∞_𝑘=2()12𝑘−1𝑘²+1²3≤−4+^∞_𝑘=12𝑘−1𝑘⁴3=−4+2𝜁(3)−𝜁(4)<1. (8)
Hence 𝐷<2. If 𝐴 has more than one element we have _{𝑎+𝑏𝑖∈𝐴}1𝑎²+𝑏²+1+√2max{𝑎,𝑏}+1⁰>1, (9) hence 𝑑>0. The result now follows from our theorem.

By a similar argument we get the second corollary.

Corollary 3. There exist finite sets 𝐴⊂ℕ[𝑖] with dim_𝐻𝔈(𝐴)>1 and there exist such sets with dim_𝐻𝔈(𝐴)<1.

Proof. Consider ^∞∞_𝑎=1𝑏=11𝑎²+𝑏²+1+√2{}≥max𝑎,𝑏+1_∞𝑘=12𝑘−12𝑘²+1+√2≥𝑘+1_∞𝑘=124+√2𝑘−12𝑘²=∞. (10)
Hence for a suitable choice of 𝐴 we have _{𝑎+𝑏𝑖∈𝐴}1𝑎²+𝑏²+1+√2max{𝑎,𝑏}+1>1. (11)
For this set 𝐴 we have 𝑑>1. On the other hand consider 𝐴={1+𝑖,2+𝑖}. We have _{𝑎+𝑏𝑖∈𝐴}1𝑎²+𝑏²<1, (12) hence 𝐷<1. The result again follows from our theorem.

We remark that it is possible to deduce the last corollaries from Theorems 1 and 2 of [9] by a few additional arguments. To obtain these results from our main theorem seems to us more transparent.

Our last corollary gives the obvious explicit upper and lower bounds following from Theorem 1.

Corollary 4. For a finite set 𝐴⊂ℕ[𝑖] with cardinality |𝐴| one has, log(|𝐴|)max_𝐴log𝑎²+𝑏²≤dim_𝐻ℭ(𝐴)≤log(|𝐴|)min_𝐴log𝑎²+𝑏²+1+√2{}.max𝑎,𝑏+1 (13)

The estimates in this corollary are of course very crude. At the end of this section we will apply Theorem 1 directly to a few examples. Let 𝐴={2+2𝑖,3+2𝑖,3+3𝑖}. The numbers 𝐷 and 𝑑 are given by 18^𝐷+113^𝐷+118^𝐷1=1,11+2√2^𝑑+117+3√2^𝑑+122+3√2^𝑑=1, (14) which implies 0.36<dim_𝐻𝔈(𝐴)<0.44, which is an acceptable estimate. If we consider values with small modulus 𝐴={1+𝑖,2+𝑖,1+2𝑖} we get 12^𝐷1+25^𝐷1=1,4+√2^𝑑1+28+2√2^𝑑=1. (15)

This gives 0.49<dim_𝐻ℭ(𝐴)<0.91, which is not very good. Let us consider one more example 𝐴={3+𝑖,2+4𝑖}. We get 110^𝐷+120^𝐷1=1,14+3√2^𝑑+125+4√2^𝑑=1 (16) and thus 0.21<dim_𝐻𝔈(𝐴)<0.27. As a last example consider 𝐴={𝑎+𝑏𝑖∣𝑎,𝑏=1⋯4}. An elementary calculation shows that Theorem 1 gives 1<dim_𝐻𝔈(𝐴)<1.33. We like to remark here that it is possible to find an algorithm using thermodynamic formalism that approximates the dimension of 𝔈(𝐴). We could apply the recent approach of Jekinsion and Pollicott [15] to infinite complex continuous fractions. This approach has the disadvantage that it is not possible to perform necessary calculations without using a computer, which would change the field of our research to computational mathematics.

3. Proof of the Result

For (𝑎,𝑏)∈ℕ² consider transformations 𝑇_𝑎,𝑏∶ℂ↦ℂ given by 𝑇_𝑎,𝑏(𝑧)=1.𝑧+𝑎+𝑏𝑖 (17)

We need three elementary lemmas concerning these transformations to apply the dimension theory of iterated functions systems to the set 𝔈(𝐴). First we restrict the maps to the open ball 𝐵_1/2(1/2)={𝑧∈𝔈∣|𝑧−1/2|<1/2}.

Lemma 5. For (𝑎,𝑏)∈ℕ² one has 𝑇_𝑎,𝑏𝐵_1/212⊂𝐵_1/212. (18)

Proof. For 𝐼(𝑧)=1/𝑧 we have 𝐼𝐵_𝑟(𝑧)=𝐵_𝑟/(|𝑧|²_−𝑟2)𝑧|𝑧|²−𝑟², (19) if |𝑧|≠𝑟. Applying the translation with 𝑎+𝑏𝑖 we obtain 𝑇_𝑎,𝑏𝐵_𝑟(𝑧)=𝐵_{𝑟/(|𝑧+𝑎+𝑏𝑖|}²_−𝑟2)𝑧+𝑎+𝑏𝑖|𝑧+𝑎+𝑏𝑖|²−𝑟² (20) if |𝑧+𝑎+𝑏𝑖|≠𝑟. Especially we get 𝑇_𝑎,𝑏𝐵_1/212=𝐵_{1/(2𝑎+2𝑎2+2𝑏}²₎1/2+𝑎−𝑏𝑖𝑎+𝑎²+𝑏². (21)
Since |(1/2)+𝑎+𝑏𝑖|>1/2 for 𝑎,𝑏>0. We have to show the distance of the center of the image to 1/2 plus the radius of the image is less or equal to 1/2. This means1/2+𝑎−𝑏𝑖𝑎+𝑎²+𝑏²−12+12𝑎+2𝑎²+2𝑏²≤12⟺𝑎²+𝑏²−𝑎−1+2𝑏𝑖²≤𝑎²+𝑏²+𝑎−1²⟺𝑎²+𝑏²−𝑎−1²+4𝑏²≤𝑎²+𝑏²+𝑎−1²⟺4𝑎+4𝑏²≤4𝑎²+𝑏²𝑎⟺1≤𝑎², (22) which is obviously true for a 𝑎∈ℕ.

Next we show that the images of the open balls 𝐵_1/2(1/2) under different 𝑇_𝑎,𝑏 are disjoint.

Lemma 6. If (𝑎₁,𝑏₁)≠(𝑎₂,𝑏₂), one has 𝑇_𝑎1,𝑏1𝐵_1/2(1/2)∩𝑇_𝑎2,𝑏2𝐵_1/2(1/2)=∅. (23)

Proof. We have to show that the distance of the balls at hand is bigger or equal to the sum of their radii, that is: 1/2+𝑎₁−𝑏₁𝑖𝑎₁+𝑎²₁+𝑏²₁−1/2+𝑎₂−𝑏₂𝑖𝑎₂+𝑎²₂+𝑏²₂≥12𝑎₁+2𝑎²₁+2𝑏²₁+12𝑎₂+2𝑎²₂+2𝑏²₂. (24)
With 𝑑₁=𝑎₁+𝑎²₁+𝑏²₁ and 𝑑₂=𝑎₂+𝑎²₂+𝑏²₂ we have to show (1/2)+𝑎₁−𝑏₁𝑖𝑑₁−(1/2)+𝑎₂−𝑏₂𝑖𝑑₂≥12𝑑₁+12𝑑₂⟺𝑑₂12+𝑎₁−𝑏₁𝑖−𝑑₁12+𝑎₂−𝑏₂𝑖²≥𝑑₁2+𝑑₂2²⟺12+𝑎₁𝑑₂−12+𝑎₂𝑑₁²+𝑏₁𝑑₂−𝑏₂𝑑₁²≥𝑑₁2+𝑑₂2²⟺12+𝑎₁²+𝑏²₁𝑑²₂1−22+𝑎₁12+𝑎₂𝑑₁𝑑₂+12+𝑎₂²+𝑏²₂𝑑²₁−2𝑏₁𝑑₂𝑏₂𝑑₁≥𝑑²₁4+𝑑₁𝑑₂2+𝑑²₂4⟺𝑑₁𝑑²₂−12+𝑎₁+𝑎₂+2𝑎₁𝑎₂𝑑₁𝑑₂+𝑑₂𝑑²₁−2𝑏₁𝑑₂𝑏₂𝑑₁≥𝑑₁𝑑₂2⟺𝑑₁+𝑑₂−1+𝑎₁+𝑎₂+2𝑎₁𝑎₂+2𝑏₁𝑏₂⟺𝑎≥0₁−𝑎₂²+𝑏₁−𝑏₂²−1≥0. (25)
This is obviously true under our assumption.

The last lemma contains estimates on the modulus of derivative of the maps on the closed ball 𝐵_1/2(1/2).

Lemma 7. For (𝑎,𝑏)∈ℕ² one has max𝑇^′_𝑎,𝑏(𝑧)𝑧∈𝐵_1/212≤1𝑎²+𝑏²,min𝑇^′_𝑎,𝑏(𝑧)𝑧∈𝐵_1/212≥1𝑎²+𝑏²+1+√2{}.max𝑎,𝑏+1 (26)

Proof. For 𝑧∈𝐵_1/2(1/2) we have 𝑇^′_𝑎,𝑏(𝑧)=−1(𝑧+𝑎+𝑏𝑖)²𝑇andhence_′𝑎,𝑏=1𝑥+𝑦𝑖(𝑥+𝑎)²+𝑦+𝑏². (27)
Now the first estimate is obvious. For the second part note that max(𝑥+𝑎)²+𝑦+𝑏²∣𝑧=𝑥+𝑖𝑦∈𝐵_1/212=max(𝑥+𝑎)²+𝑦+𝑏²∣1𝑥−2²+𝑦²≤14,𝑥,𝑦∈ℝ=𝑎²+𝑏²+max𝑥+2𝑥𝑎+2𝑦𝑏∣𝑦²≤𝑥−𝑥²,𝑥∈[0,1],𝑦∈−12,12≤1+𝑎²+𝑏²+2max𝑥𝑎+𝑦𝑏∣𝑦²≤𝑥−𝑥²,𝑥∈[0,1],𝑦∈−12,12≤1+𝑎²+𝑏²+2max{𝑎,𝑏}max𝑥+𝑦∣𝑦²≤𝑥−𝑥²,𝑥∈[0,1],𝑦∈−12,12≤1+𝑎²+𝑏²+1+√2max{𝑎,𝑏} (28) using the approach of Lagrange in the last estimate. This implies the result.

Given a finite set 𝐴⊂ℕ[𝑖] consider the iterated function system (IFS) in the sense of Hutchinson [16]: 𝐵_1/212,𝑇_𝑎,𝑏∣𝑎+𝑏𝑖∈𝐴. (29)

By Lemma 5 this IFS is well defined with attractor ℭ(𝐴), that is: ℭ(𝐴)=_{𝑎+𝑏𝑖∈𝐴}𝑇_𝑎,𝑏(ℭ(𝐴)). (30)

By Lemma 6 the IFS fulfills the open set condition, first introduced by Moran [11]. Moreover by Lemma 7 we have 𝑧₁−𝑧₂𝑎²+𝑏²+1+√2{}≤𝑇max𝑎,𝑏+1_𝑎,𝑏𝑧₁−𝑇_𝑎,𝑏𝑧₂≤𝑧₁−𝑧₂𝑎²+𝑏² (31) for all 𝑧₁,𝑧₂∈𝐵_1/2(1/2) and all 𝑎+𝑏𝑖∈𝐴. Now Theorem 1 is a direct application of Theorem 8.8 of Falconer [17], a well-know result in the dimension theory of IFS, which goes back to Moran [11].