## 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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