×

zbMATH — the first resource for mathematics

On one fractal property of the Minkowski function. (English) Zbl 1387.28013
Summary: The article is devoted to fractal properties of the singular Minkowski function. It is proved that this function does not belong to the class of DP-transformations, i.e., the Minkowski function does not preserve the Hausdorff-Besicovitch dimension.

MSC:
28A80 Fractals
11K50 Metric theory of continued fractions
11K55 Metric theory of other algorithms and expansions; measure and Hausdorff dimension
26A30 Singular functions, Cantor functions, functions with other special properties
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Good, GT, The fractional dimensional theory of continued fractions, Proc. Camb. Phil. Soc., 37, 199-228, (1941) · Zbl 0061.09408
[2] Hensley, D, Continued fraction Cantor sets, Hausdorff dimension and functional analysis, J. Number Theory, 40, 336-358, (1992) · Zbl 0745.28005
[3] Hirst, KE, Fractional dimension theory of continued fractions, Q. J. Math., 21, 29-35, (1970) · Zbl 0191.33301
[4] Jarnik, V, Zur metrichen theorie den diophantischen approximationen, Recl. Math. Mosc., 36, 371-382, (1929) · JFM 55.0719.01
[5] Minkowski, H.: Zur Geometrie der Zahlen. In: Minkowski, H. (ed.) Gesammeine Abhandlungen, Band 2, pp. 50-51. Druck und Verlag von B. G. Teubner, Leipzig und Berlin (1911) · Zbl 0745.28005
[6] Serbenyuk S. O.: Preserving the Hausdorff-Besicovitch dimension by monotonic singular distribution functions. In: Second interuniversity scientific conference on mathematics and physics for young scientists: abstracts, pp. 106-107. Institute of Mathematics of NAS of Ukraine, Kyiv (2011). https://www.researchgate.net/publication/301637057 (in Ukrainian)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.