固定强度球面设计
Fixed-strength spherical designs
AI总结：
本文研究了固定t和维度趋于无穷大的情况下球面设计的最小大小，通过高斯设计构造更小的设计并确定了带符号t-设计的精确数量级。
摘要：
球面t-设计是单位球上的有限子集X，使得所有次数不超过t的多项式在X上的平均值等于其在整个球面上的平均值。确定球面设计的最小可能大小，尤其是当t趋于无穷大时在固定维度中的情况，已成为几十年来的重要研究课题。本文提出了关于互补渐进行为的研究结果，其中t固定而维度趋于无穷大。本文的主要结果是（1）通过显式连接到高斯设计来构造更小的球面设计；（2）确定最小大小的带符号t-设计的精确数量级，这比典型的自由度启发法预测的要小得多。我们还建立了一种将球面设计在不同维度之间“投影”的方法，证明了多种关于近似设计的结果，并构造了新的t-wise独立子集{1,2,…,q}^d，这可能有独立的兴趣。为了达到这些结果，我们结合了代数、几何、概率、表示论和优化技术。
英文摘要：
A spherical $t$-design is a finite subset $X$ of the unit sphere such that every polynomial of degree at most $t$ has the same average over $X$ as it does over the entire sphere. Determining the minimum possible size of spherical designs, especially in a fixed dimension as $t \to \infty$, has been an important research topic for several decades. This paper presents results on the complementary asymptotic regime, where $t$ is fixed and the dimension tends to infinity. The main results in this paper are (1) a construction of smaller spherical designs via an explicit connection to Gaussian designs and (2) the exact order of magnitude of minimal-size signed $t$-designs, which is significantly smaller than predicted by a typical degrees-of-freedom heuristic. We also establish a method to ``project'' spherical designs between dimensions, prove a variety of results on approximate designs, and construct new $t$-wise independent subsets of $\{1,2,\dots,q\}^d$ which may be of independent interest. To achieve these results, we combine techniques from algebra, geometry, probability, representation theory, and optimization.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
