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

TL;DR

This paper proves the existence of asymptotically optimal $t$-design curves on the 3-sphere ($S^3$), with lengths proportional to $t^2$, extending previous results from the 2-sphere to higher dimensions.

Contribution

It establishes the existence of simple $t$-design curves on $S^3$ with length proportional to $t^2$, solving an open problem for the three-dimensional sphere.

Findings

Existence of $t$-design curves on $S^3$ with length $Ct^2$.

Construction of simple $t$-design curves on $S^3$.

Extension of asymptotic optimality results from $S^2$ to $S^3$.

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 at most $t$ evaluates to the average of this polynomial on the sphere. These authors posed the problem of proving that there exist sequences $(\gamma_t)_{t=0}^\infty$ of $t$-design curves on $S^d$ of asymptotically optimal length $\ell(\gamma_t)\asymp t^{d-1}$ as $t\to\infty$ and solved this problem for $d=2$. This work solves the problem for $d=3$ by proving that there exists a constant $\mathscr C>0$ such that for any $C\geq\mathscr C$ and $t\in\Bbb N_+$, there exists a simple $t$-design curve on $S^3$ of length $Ct^2$.