Inscribed rectangles in a smooth Jordan curve attain at least one third of all aspect ratios

TLDR: In this paper, it was shown that for every smooth Jordan curve, the Lebesgue measure of a set of disjoint Mobius strips bounding a torus knot in the solid torus times an interval is at least 1/3.

Abstract: We prove that for every smooth Jordan curve $\gamma$, if $X$ is the set of all $r \in [0,1]$ so that there is an inscribed rectangle in $\gamma$ of aspect ratio $\tan(r\cdot \pi/4)$, then the Lebesgue measure of $X$ is at least $1/3$. To do this, we study disjoint Mobius strips bounding a $(2n,n)$-torus link in the solid torus times an interval. We prove that any such set of Mobius strips can be equipped with a natural total ordering. We then combine this total ordering with some additive combinatorics to prove that $1/3$ is a sharp lower bound on the probability that a Mobius strip bounding the $(2,1)$-torus knot in the solid torus times an interval will intersect its rotation by a uniformly random angle.