On the packing measure of slices of self-similar sets

Abstract

Let $K\subset {\mathbb{R}}^2$ be a rotation and reflection free self-similar set satisfying the strong separation condition, with dimension $\dim K=s>1$. Intersecting $K$ with translates of a fixed line, one can study the $(s-1)$-dimensional Hausdorff and packing measures of the generic non-empty line sections. In a recent article, T. Kempton gave a necessary and sufficient condition for the Hausdorff measures of the sections to be positive. In this paper, I consider the packing measures: it turns out that the generic section has infinite $(s-1)$-dimensional packing measure under relatively mild assumptions.

10 pages, 2 figures. v3: incorporated referee suggestions, including the removal of the equicontractivity assumption from the main theorem, to appear in J. Fractal Geom

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