Asymptotically optimal $t$-design curves on $S^3$

Abstract: A $\textit{spherical $t$-design curve}$ was defined by Ehler and Gr\"{o}chenig to be a continuous, piecewise smooth, closed curve on the sphere with finitely many self-intersections whose associated line integral applied to any polynomial of degree $t$ or less evaluates to the average of this polynomial on the sphere. These authors posed the problem of proving that there exist sequences ${\gamma_t}{t\in\Bbb N}$ of $t$-design curves on $S^d$ of asymptotically optimal length $\ell(\gamma_t)=\Theta(t^{d-1})$ as $t\to\infty$ and solved this problem for $d=2$. This work solves the problem for $d=3$ by proving existence of a constant $\mathcal C>0$ such that for any $C>\mathcal C$ and $t\in\Bbb N+$, there exists a $t$-design curve with no self-intersections on $S^3$ of length $Ct^2$.