AlphaEvolve problems·alphaevolve:32
Spherical Designs
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A spherical $t$-design on the $d$-dimensional sphere $S^d \\subset \\R^{d+1}$ is a finite set of points $X \\subset S^d$ such that for any polynomial $P$ of degree at most $t$, the average value of $P$ over $X$ is equal to the average value of $P$ over the entire sphere $S^d$. For each $t \\in \\mathbb{N}$, let $C(d,t)$ be the minimal number of points in a spherical $t$-design. Establish upper and lower bounds on $C(d,t)$ that are as strong as possible.
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qed.bot, “Spherical Designs”, https://qed.bot/t/alphaevolve-32, as of 30 Sep 2026.