Algebra Seminar talk

Paul Großkopf
Intersecting the Twin Dragon with rational lines

The Twin Dragon is a certain well known compact, connected subset of the plane closely related to the dragon curve. It appears in the radix representation of complex numbers in base -1+i and its boundary is a fractal with Hausdorff dimension 1.5236… . It is known that the intersection of a (Borel) fractal in $\mathbb R^2$ with a straight line always reduces its Hausdorff dimension by 1, except for a Lebesgue measure null set of lines (where the Lebesgue measure is understood to measure a set of natural parameters for these lines).

Although this theorem applies to the Twin Dragon, all intersections for which the Hausdorff measure is known lie in the exceptional null set. Following techniques of Akiyama and Scheicher using Büchi automata it is possible to analyze further rational lines.