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The Hausdorff dimension of general Sierpiński carpets. (English) Zbl 0539.28003

The simplest examples of fractional-dimensional sets which are self-affine but not self-similar are ”Sierpiński carpets” in the plane. An example is the set of points \((x,y)\) in the unit square such that zero appears in the base 2 representation of \(x\) iff a zero occurs in the corresponding place of the base 3 expansion of y. We determine the Hausdorff dimension of such sets, and observe that it agrees with the metric or capacity dimension only in exceptional cases. As a byproduct we obtain the Hausdorff dimension of a certain continuous self-affine curve constructed by Hironaka.

MSC:

28A75 Length, area, volume, other geometric measure theory
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References:

[1] Hironaka Heisuke no Suugaku Kyositsu (1980)
[2] DOI: 10.1017/S0305004100039049
[3] J. Reine Angew. Math 246 pp 46– (1971)
[4] Probability Theory (1963) · Zbl 0132.00104
[5] Probability Measures on Metric Spaces (1967) · Zbl 0153.19101
[6] Fractals (1977)
[7] Comptes Rendus 162 pp 629– (1916)
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