The boundary of a smooth set has full Hausdorff dimension. (English) Zbl 1077.28008

Summary: We prove that if the restriction of the Lebesgue measure to a set \(A\subset [0,1]\) with \(0<|A|<1\) is a smooth measure, then the boundary of \(A\) must have full Hausdorff dimension.


28A80 Fractals
28A78 Hausdorff and packing measures
Full Text: DOI


[1] Carleson, L., On mappings conformal at the boundary, J. Anal. Math., 19, 1-13 (1967) · Zbl 0186.13701
[2] Duren, P. L.; Shapiro, H. S.; Shields, A., Singular measures and domains not of Smirnov type, Duke Math. J., 33, 247-254 (1966) · Zbl 0174.37501
[3] Kahane, J. P., Trois notes sur les ensembles parfait linéaires, Enseign. Math., 15, 185-192 (1969) · Zbl 0175.33902
[4] Piranian, G., Two monotonic, singular, uniformly almost smooth functions, Duke Math. J., 33, 255-262 (1966) · Zbl 0143.07405
[5] Sarason, D. E., Blaschke products in \(B0\), (Havin, V. P.; Hruščëv, S. V.; Nikol’skii, N. K., Linear and Complex Analysis Problem Book. Linear and Complex Analysis Problem Book, Lecture Notes in Math., vol. 1043 (1984), Springer-Verlag: Springer-Verlag Berlin)
[6] Shapiro, H. S., Monotonic singular functions of high smoothness, Michigan Math. J., 15, 265-275 (1968) · Zbl 0165.06904
[7] Smith, W., Inner functions in the hyperbolic little Bloch class, Michigan Math. J., 45, 1, 103-114 (1998) · Zbl 0976.30018
[8] Zygmund, A., Trigonometric Series (1959), Cambridge Univ. Press: Cambridge Univ. Press Cambridge, UK · JFM 58.0280.01
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.