Jordan curves inscribe a positive measure of rectangles

Abstract

Suppose that $\gamma \subset \mathbb{C}$ is a Jordan curve of diameter $2R$ which encloses a region of area $A$. We prove that there exists a subset $I \subset (0,\pi)$ of measure at least $A/R^2$ such that if $\theta \in I$, then there exist four points on $\gamma$ at the vertices of a rectangle whose diagonals meet at angle $\theta$.