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Journal of Mathematics
Volume 2013 (2013), Article ID 754134, 5 pages
http://dx.doi.org/10.1155/2013/754134
Research Article

Dimension Estimates for Certain Sets of Infinite Complex Continued Fractions

Institut für Mathematik, Technical University Clausthal, Reitstallweg 9, 38640 Goslar, Germany

Received 13 August 2012; Accepted 2 October 2012

Academic Editor: Stefan Siegmund

Copyright © 2013 J. Neunhäuserer. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 [37]. 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 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 . 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 for of Gaussian integers we define the infinite complex continued fraction by

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

We consider the set of all infinite continued fractions having fractional entries coming from :

Obviously the set 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 -dimensional Hausdorff measure of a set is

The Hausdorff dimension of is given by

Now we are able to state our main result on .

Theorem 1. For a finite set let be the unique real numbers fulfilling
We have

Unsing an additionla argument this theorem has the following corollary.

Corollary 2. For all finite sets one has ; on the other hand if has more than one element.

Proof. Consider
Hence . If has more than one element we have hence . The result now follows from our theorem.

By a similar argument we get the second corollary.

Corollary 3. There exist finite sets with and there exist such sets with .

Proof. Consider
Hence for a suitable choice of we have
For this set we have . On the other hand consider . We have hence . 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 with cardinality one has,

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 . The numbers and are given by which implies , which is an acceptable estimate. If we consider values with small modulus we get

This gives , which is not very good. Let us consider one more example . We get and thus . As a last example consider . An elementary calculation shows that Theorem 1 gives . We like to remark here that it is possible to find an algorithm using thermodynamic formalism that approximates the dimension of . 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 consider transformations given by

We need three elementary lemmas concerning these transformations to apply the dimension theory of iterated functions systems to the set . First we restrict the maps to the open ball .

Lemma 5. For one has

Proof. For we have if . Applying the translation with we obtain if . Especially we get
Since for . We have to show the distance of the center of the image to plus the radius of the image is less or equal to . This means which is obviously true for a .

Next we show that the images of the open balls under different are disjoint.

Lemma 6. If , one has

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:
With and we have to show
This is obviously true under our assumption.

The last lemma contains estimates on the modulus of derivative of the maps on the closed ball .

Lemma 7. For one has

Proof. For we have
Now the first estimate is obvious. For the second part note that using the approach of Lagrange in the last estimate. This implies the result.

Given a finite set consider the iterated function system (IFS) in the sense of Hutchinson [16]:

By Lemma 5 this IFS is well defined with attractor , that is:

By Lemma 6 the IFS fulfills the open set condition, first introduced by Moran [11]. Moreover by Lemma 7 we have for all and all . 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].

References

  1. J. Havil, Gamma: Exploring Euler's Constant, Princeton University Press, Princeton, NJ, USA, 2003. View at MathSciNet
  2. V. Jarnik, “Diophantische Approximationen und Hausdorffsches Mass,” Rec. Math. Soc. Moscou, vol. 36, pp. 371–382, 1929.
  3. I. J. Good, “The fractional dimension of continued fractions,” Proceedings of the Cambridge Philosophical Society, vol. 37, pp. 199–208, 1941. View at Zentralblatt MATH · View at MathSciNet
  4. R. T. Bumby, “Hausdorff dimension of Cantor sets,” Journal für die reine und angewandte Mathematik, vol. 331, pp. 192–206, 1982. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  5. R. T. Bumby, Hausdorff Dimension of Sets Arising in Number Theory, vol. 1135 of Lecture Notes in Mathematics, Springer, New York, NY, USA, 1985. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  6. D. Hensley, “The Hausdorff dimensions of some continued fraction cantor sets,” Journal of Number Theory, vol. 33, no. 2, pp. 182–198, 1989. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  7. D. Hensley, “A polynomial time algorithm for the Hausdorff dimension of continued fraction Cantor sets,” Journal of Number Theory, vol. 58, no. 1, pp. 9–45, 1996. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  8. O. Jenkinson and M. Pollicott, “Computing the dimension of dynamically defined sets: E2 and bounded continued fractions,” Ergodic Theory and Dynamical Systems, vol. 21, no. 5, pp. 1429–1445, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  9. R. J. Gardner and R. D. Mauldin, “On the Hausdorff dimension of a set of complex contiuned fractions,” Illinois Journal of Mathematics, vol. 27, no. 2, pp. 334–345, 1983. View at Zentralblatt MATH · View at MathSciNet
  10. R. D. Mauldin and M. Urbański, “Dimensions and measures in infinite iterated function systems,” Proceedings of the London Mathematical Society, vol. 73, no. 1, pp. 105–154, 1996. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  11. P. Morna, “Additiv funktions of Intervalls and Hausdorff dimension,” Proceedings of the Cambridge Philosophical Society, vol. 42, pp. 15–23, 1946. View at Zentralblatt MATH · View at MathSciNet
  12. A. Hurwitz, “Uber die Entwicklung Complexer Grössen in Kettenbruche,” Acta Mathematica, vol. 11, pp. 187–200, 1895. View at Publisher · View at Google Scholar · View at MathSciNet
  13. K. Falconer, Fractal Geometry—Mathematical Foundations and Applications, Wiley, New York, NY, USA, 1990. View at MathSciNet
  14. Y. A. Pesin, Dimension Theory in Dynamical Systems—Contemplary Views and Applications, University of Chicago Press, 1997. View at MathSciNet
  15. O. Jenkinson and M. Pollicott, “Calculating Hausdorff dimension of Julia sets and Kleinian limit sets,” American Journal of Mathematics, vol. 124, no. 3, pp. 495–545, 2002. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  16. J. E. Hutchinson, “Fractals and self-similarity,” Indiana University Mathematics Journal, vol. 30, pp. 271–280, 1981. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  17. K. Falconer, Techniques in Fractal Geometry, Wiley, New York, NY, USA, 1997. View at MathSciNet