Asymptotically short generalizations of $t$-design curves

Ehler and Gröchenig posed the question of finding $t$-design curves ${\gamma }_t$$\text{unicode}x2013$curves whose associated line integrals exactly average all degree at most $t$ polynomials$\text{unicode}x2013$on $S^d$ of asymptotically optimal arc length $\mathrm{\ell }({\gamma }_t)\asymp t^{d-1}$ as $t\to \mathrm{\infty }$. This work investigates analogues of this question for $\mathit{\text{weighted}}$ and \textit{\varepsilon_t$-approximate$t-design curves}, proving existence of such curves ${\gamma }_t$ on $S^d$ of arc length $\mathrm{\ell }({\gamma }_t)\asymp t^{d-1}$ as $t\to \mathrm{\infty }$ for all $d\in {\mathbb{N}}_{+}$ in the weighted setting (in which case such curves are asymptotically optimal) and all odd $d\in {\mathbb{N}}_{+}$ in the approximate setting (where we have ${\epsilon }_t\asymp 1\mathrm{/}t$ as $t\to \mathrm{\infty }$). Formulas for such weighted $t$-design curves for $d\in \{2,3\}$ are presented.