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UNIQUE EXPANSION OF POINTS OF A CLASS OF SELF-SIMILAR SETS WITH OVERLAPS

Part of: Approximations and expansions

Published online by Cambridge University Press:  25 April 2012

Yuru Zou ,
Jian Lu  and
Wenxia Li
Yuru Zou
Affiliation:
College of Mathematics and Computational Science, Shenzhen University, Shenzhen 518060, PR China (email: [email protected])
Jian Lu
Affiliation:
College of Mathematics and Computational Science, Shenzhen University, Shenzhen 518060, PR China (email: [email protected])
Wenxia Li*
Affiliation:
Department of Mathematics, East China Normal University, Shanghai 200241, PR China (email: [email protected])
*
*Wenxia Li is the corresponding author.
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Abstract

For q>1, the set Fq of real numbers which can be expanded in base q with respect to the digit set {0,1,q} is just a self-similar set with overlaps. We consider the subset of Fq whose elements have a unique expansion and calculate its Hausdorff dimension for the case where .

MSC classification

Secondary: 41A99: None of the above, but in this section
Type
Research Article
Information
Mathematika , Volume 58 , Issue 2 , July 2012 , pp. 371 - 388
Copyright
Copyright © University College London 2012

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References

[1]Daróczy, Z. and Kátai, I., On the structure of univoque numbers. Publ. Math. Debrecen 46 (1995), 385–408.CrossRefGoogle Scholar
[2]Edgar, G. A., A fractal puzzle. Math. Intelligencer 13 (1991), 44–50.CrossRefGoogle Scholar
[3]Erdős, P., Horváth, M. and Joó, I., On the uniqueness of the expansions 1=∑ ∞i=1q −n i. Acta Math. Hungar. 58(3–4) (1991), 333–342.Google Scholar
[4]Erdős, P., Joó, I. and Komornik, V., Characterization of the unique expansions 1=∑ ∞i=1q −n i and related problems. Bull. Soc. Math. France 118 (1990), 377–390.Google Scholar
[5]Falconer, K. J., Fractal geometry. In Mathematical Foundations and Applications, John Wiley & Sons (Chichester, 1990).Google Scholar
[6]Glendinning, P. and Sidorov, N., Unique representations of real numbers in non-integer bases. Math. Res. Lett. 8 (2001), 535–543.CrossRefGoogle Scholar
[7]Hutchinson, J. E., Fractal and self-similarity. Indiana Univ. Math. J. 30 (1981), 713–747.Google Scholar
[8]Komornik, V. and Loreti, P., Unique developments in non-integer bases. Amer. Math. Monthly 105 (1998), 936–939.Google Scholar
[9]Komornik, V. and Loreti, P., On the expansions in non-integer bases. Rend. Mat. Appl. 19 (2000), 615–634.Google Scholar
[10]Komornik, V. and Loreti, P., On the topological structure of univoque sets. J. Number Theory 122 (2007), 157–183.CrossRefGoogle Scholar
[11]Lalley, S. P., β-expansions with deleted digits for Pisot numbers. Trans. Amer. Math. Soc. 349 (1997), 4355–4365.Google Scholar
[12]Lau, K. S. and Ngai, S. M., A generalized finite type condition for iterated function systems. Adv. Math. 208 (2007), 647–671.Google Scholar
[13]Mauldin, R. and Williams, S., Hausdorff dimension in graph directed constructions. Trans. Amer. Math. Soc. 309 (1988), 811–829.Google Scholar
[14]Ngai, S. M. and Wang, Y., Hausdorff dimension of self-similar sets with overlaps. J. Lond. Math. Soc. 63 (2001), 655–672.CrossRefGoogle Scholar
[15]Pedicini, M., Greedy expansions and sets with deleted digits. Theoret. Comput. Sci. 332 (2005), 313–336.CrossRefGoogle Scholar
[16]Rao, H. and Wen, Z. Y., A class of self-similar fractals with overlap structrue. Adv. App. Math. 20 (1998), 50–72.Google Scholar
[17]Schief, A., Separation properties for self-similar sets. Proc. Amer. Math. Soc. 122 (1994), 111–115.CrossRefGoogle Scholar
[18]Strichartz, R. S. and Wang, Y., Geometry of self-similar tiles I. Indiana Univ. Math. J. 48 (1999), 1–23.Google Scholar
[19]Vries, M. D. and Komornik, V., Unique expansions of real numbers. Adv. Math. 221 (2009), 390–427.CrossRefGoogle Scholar
[20]Zerner, M. P. W., Weak separation properties for self-similar sets. Proc. Amer. Math. Soc. 124 (1996), 3529–3539.CrossRefGoogle Scholar