Dimension functions for fractal sets associated to series
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- by Manuel Morán PDF
- Proc. Amer. Math. Soc. 120 (1994), 749-754 Request permission
Abstract:
In this article we analyze suitable dimension functions for the measure of fractal sets associated to certain absolutely convergent series of vectors of ${\mathbb {R}^n}$. The complex binomial series provides an example of a family of fractals, all of them with the same Hausdorff dimension but with a totally ordered family of suitable dimension functions indexed by a parameter with range in a real interval. We also show a construction, based on sets associated to series, to obtain fractal sets with given dimension function.References
- C. A. Rogers, Hausdorff measures, Cambridge University Press, London-New York, 1970. MR 0281862
- M. Morán, Fractal series, Mathematika 36 (1989), no. 2, 334–348 (1990). MR 1045792, DOI 10.1112/S0025579300013176
- Børge Jessen and Aurel Wintner, Distribution functions and the Riemann zeta function, Trans. Amer. Math. Soc. 38 (1935), no. 1, 48–88. MR 1501802, DOI 10.1090/S0002-9947-1935-1501802-5
- Paul Erdös, On the smoothness properties of a family of Bernoulli convolutions, Amer. J. Math. 62 (1940), 180–186. MR 858, DOI 10.2307/2371446
- Adriano M. Garsia, Arithmetic properties of Bernoulli convolutions, Trans. Amer. Math. Soc. 102 (1962), 409–432. MR 137961, DOI 10.1090/S0002-9947-1962-0137961-5
- Kenneth Falconer, Fractal geometry, John Wiley & Sons, Ltd., Chichester, 1990. Mathematical foundations and applications. MR 1102677
Additional Information
- © Copyright 1994 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 120 (1994), 749-754
- MSC: Primary 28A78; Secondary 28A80
- DOI: https://doi.org/10.1090/S0002-9939-1994-1186131-7
- MathSciNet review: 1186131