Algebra Seminar talk
2020-01-10
Paul Großkopf
Intersecting the Twin Dragon with rational lines
Abstract:
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.