Asymptotically short generalizations of $t$-design curves

Ehler and Gr\"{o}chenig posed the question of finding $t$-design curves $\gamma_t$$\unicode{x2013}$curves whose associated line integrals exactly average all degree at most $t$ polynomials$\unicode{x2013}$on $S^d$ of asymptotically optimal arc length $\ell(\gamma_t)\asymp t^{d-1}$ as $t\to\infty$. This work investigates analogues of this question for $\textit{weighted}$ and $\textit{$\varepsilon_t$-approximate $t$-design curves}$, proving existence of such curves $\gamma_t$ on $S^d$ of arc length $\ell(\gamma_t)\asymp t^{d-1}$ as $t\to\infty$ for all $d\in\Bbb N_+$ in the weighted setting (in which case such curves are asymptotically optimal) and all odd $d\in\Bbb N_+$ in the approximate setting (where we have $\varepsilon_t\asymp1/t$ as $t\to\infty$). Formulas for such weighted $t$-design curves for $d\in\{2,3\}$ are presented.