Dimension of slices of Sierpiński-like carpets

Abstract

We investigate the dimension of intersections of the Sierpiński-like carpets with lines. We show a sufficient condition that for a fixed rational slope the dimension of almost every intersection with respect to the natural measure is strictly greater than $s–1$, and almost every intersection with respect to the Lebesgue measure is strictly less than $s–1$, where s is the Hausdorff dimension of the carpet. Moreover, we give partial multifractal spectra for the Hausdorff and packing dimension of slices.