Asymptotically short generalizations of $t$-design curves

TL;DR

The paper addresses the problem of finding asymptotically short generalizations of spherical $t$-designs by introducing weighted and $\varepsilon_t$-approximate $t$-design curves. It develops a framework that builds such curves from complex projective design sets, using a fiber-connecting construction to transfer averaging properties from $\mathbb{CP}^n$ to spheres $S^{2n+1}$ and $S^{2n}$, and proves that the resulting curves achieve arc-length scales $ ext{length}(\gamma_t)\asymp t^{d-1}$ with asymptotically optimal constants for weighted curves (all $d$) and for odd-dimensional spheres in the approximate setting, with explicit constructions in low dimensions ($d=2,3$). The main contributions include a rigorous construction of weighted $t$-design curves for all $d>1$ with explicit formulas in $d=2,3$, and a complete asymptotic theory for approximate $t$-design curves on both odd and even spheres, showing $ ext{length}(\gamma_t)\asymp t^{d-1}$ and $\boldsymbol{\varepsilon}_t\asymp 1/t$ in the appropriate dimensions. The work offers a versatile bridge between CP-design theory and spherical designs, enabling potential extensions to hybrid designs and design-submanifolds with applications to numerical integration, data analysis on spheres, and experimental design.

Abstract

Ehler and Gröchenig posed the question of finding $t$-design curves $γ_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 $\ell(γ_t)\asymp t^{d-1}$ 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 $γ_t$ on $S^d$ of arc length $\ell(γ_t)\asymp t^{d-1}$ 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.

Figures (6)

• Figure 1: Images of the weighted 1-design curve $w_{V_2,I_3}$ (left) and the weighted 3-design curve $w_{V_4,I_3}$ (right) on $S^2$ respectively resulting from the construction of Theorem \ref{['thm:weightconst']} applied to a 1-design set (antipodal points $V_2$) and 3-design set (the vertices $V_4$ of a 4-gon) on $S^1$.
• Figure 2: Images of a weighted 1-design curve (left) and a weighted 3-design curve (right) on $S^3\cong\Bbb R^3\cup\{\infty\}$ respectively resulting from the construction of Theorem \ref{['thm:weightconst']} applied to a 1-design set (antipodal points) and 3-design set (the vertices of an octahedron) on $S^2$.
• Figure 3: The preimage $\Pi^{-1}(O)\subset S^3\cong\Bbb R^3\cup\{\infty\}$ under $\Pi$ of the 3-design set on $\Bbb{CP}^1$ corresponding to the vertices $O$ of an octahedron under the association $\Bbb{CP}^1\cong S^2$BannaiBannai09. The average of any degree at most 7 polynomial on $\Pi^{-1}(O)$ equals the average of this polynomial on $S^3$.
• Figure 4: Making the tree $\mathcal{T}_O$ as in Section \ref{['sec:const']} associated to the vertices $O$ of an octahedron on $\Bbb{CP}^1\cong S^2$.
• Figure 5: Connecting the disjoint circles comprising $\Pi^{-1}(O)$ to form a path-connected figure $\Gamma_O$ in accordance with the tree $\mathcal{T}_O$ as in Figure \ref{['fig:MakingGraph']}. This is the image of a $((5\pi+\delta)/(12\pi-\delta),12\pi/(17\pi-\delta))$-approximate 7-design cycle, with $\delta$ as in Theorem \ref{['thm:const']}.

Theorems & Definitions (29)

• Definition 1.1
• Definition 1.2
• Theorem 1.3: Main theorem on weighted $t$-design curves
• Theorem 1.4
• Theorem 1.5
• Theorem 1.6: Main theorem on approximate $t$-design curves
• Theorem 1.7
• Theorem 1.8
• Definition 2.1
• proof : Proof of Theorem \ref{['thm:weightconst']}