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Dimension prints of fractal sets. (English) Zbl 0799.28006

Dimension prints were developed in 1988 to distinguish between different fractal sets in Euclidean spaces having the same Hausdorff dimension but with very different geometric characteristics. In this paper we compute the dimension prints of some fractal sets, including generalized Cantor sets on the unit circle \(S^ 1\) in \({\mathfrak R}^ 2\) and the graph of generalized Lebesgue functions also in \({\mathfrak R}^ 2\). In this second case we show that the dimension print for the graphs of the Lebesgue functions can approach the maximal dimension print of a set of Hausdorff dimension 1. We study the dimension prints of the Cartesian product of linear Borel sets and obtain the exact dimension print when each linear set has positive measure in its dimension and the dimension of the Cartesian product is the sum of the dimensions of the factors.

MSC:

28A78 Hausdorff and packing measures
28A80 Fractals
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References:

[1] DOI: 10.1112/jlms/s1-27.3.295 · Zbl 0046.28203
[2] Rogers, Mathematika 35 pp 1– (1988)
[3] Rogers, Hausdorff Measures (1970)
[4] DOI: 10.1112/jlms/s1-20.2.110 · Zbl 0063.00354
[5] Falconer, Fractal Geometry (1990)
[6] Falconer, The Geometry of Fractal Sets (1985) · Zbl 0587.28004
[7] DOI: 10.1017/S0305004100029236
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