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An exceptional set for Hausdorff dimension

Published online by Cambridge University Press:  26 February 2010

R. Kaufman
R. Kaufman
Affiliation:
University of Illinois, Urbana, Illinois, U.S.A.
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Let D be a compact set in the plane, t a real number, and Dt the linear set {x + ty|x + iy ε D}. We are interested in the Hausdorff dimensions of D and Dt, and assume that dim D = d ≤ 1. A number t is “exceptional” if dim Dt < d; the exceptional numbers form a Borel set of d-dimensional measure zero [3]. (Marstrand [4, p. 268] proves a similar conclusion for the Lebesgue measure of the exceptional set.) In this note we exhibit a planar set D of Hausdorff dimension d and a linear set E for which dim Dt ≤ r < d for every t in E, but dim E > 0. The method does not come very close to the probable truth, that the set E can have dimension r. But perhaps the result can be improved by more subtle calculation, for example, as in Jarník [2].

Type
Research Article
Information
Mathematika , Volume 16 , Issue 1 , June 1969 , pp. 57 - 58
Copyright
Copyright © University College London 1969

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References

1.Eggleston, H. G., “Sets of fractional dimensions…”, Proc. London Math. Soc. (2), 54 (1951), 42–93.Google Scholar
2.Jarník, V., “Über die simultanen diophantischen Approximationen”, Math. Z., 33 (1931), 505–543.CrossRefGoogle Scholar
3.Kaufman, R., “Dimension of Projections”, Mathematika, 15 (1968), 153–155.CrossRefGoogle Scholar
4.Marstrand, J. M., “Some fundamental geometrical properties…”, Proc. London Math. Soc. (3), 4 (1954), 257–302.CrossRefGoogle Scholar