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The Hausdorff dimension of some special plane sets. (English) Zbl 0813.28001

The paper gives a self-contained construction of a compact plane set \(T\) such that every horizontal or vertical line intersects \(T\) in at most one point, but for \(0< \alpha< 2\) the \(\alpha\)-dimensional Hausdorff measure of \(T\) is infinite. The construction is flexible enough to allow an estimation of the Hausdorff measure of \(T\) for certain measure functions, from below and above. Because the author is deceased during editorial process, some editorial comments at the end of the paper relate this construction to more general results due to Mattila.

MSC:

28A05 Classes of sets (Borel fields, \(\sigma\)-rings, etc.), measurable sets, Suslin sets, analytic sets
28A78 Hausdorff and packing measures