Slicing the Sierpi\'nski gasket
Bal\'azs B\'ar\'any, Andrew Ferguson, K\'aroly Simon
TL;DR
This paper studies the intersection properties of the Sierpiński gasket with lines, identifying exceptional angles and analyzing the multifractal structure of slices with specific dimensions.
Contribution
It characterizes a dense set of exceptional angles for Marstrand's theorem and provides a multifractal analysis of the slices of the gasket.
Findings
Identifies a countable, dense set of exceptional angles.
Provides a multifractal spectrum for the slices.
Analyzes the dimension of intersections with lines.
Abstract
We investigate the dimension of intersections of the Sierpi\'nski gasket with lines. Our first main result describes a countable, dense set of angles that are exceptional for Marstrand's theorem. We then provide a multifractal analysis for the set of points in the projection for which the associated slice has a prescribed dimension.
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Taxonomy
TopicsMathematical Dynamics and Fractals · Digital Image Processing Techniques · Advanced Topology and Set Theory