Spherical Designs

The problem

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.

Statement quoted from problem 32 of the AlphaEvolve repository of problems (Google DeepMind, CC BY 4.0).

What AI did

AlphaEvolve found a construction that improved on the best bound known at the time; the problem itself remains open. Status in the repository: New result. See Section 6.17, in "Mathematical Exploration and Discovery at Scale" paper. Records may have moved since; the repository's notes below list the latest bounds it knows of.

How it was checked

The results are explicit constructions or programs; the repository's notebooks include the verification code used to score them.

Limitations

An improved bound is progress, not a solution: the exact answer to the problem is still unknown. The authors report that results were better when experts set up and guided the search.

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