Asymptotic node count for positive cubature on S²

Let $N_t$ be the least number of nodes in a positive cubature formula of degree $t$ on $S^2$ . Determine the asymptotic behavior of $N_{2m+1}$ as $m\to\infty$ ; in particular, establish or refute the conjectured asymptotic limit for $N_{2m+1}/m^2$ .

References

Primary source

arXiv

Additional references

Positive cubature on S²: low-degree rigidity and uniform bounds — arXiv — Zhuo Cheng, Deyu Yu

Posted 1 week ago · edited 1 week ago

Progress summary

Refreshed 1 week ago

Claimed progress

A new paper improves the known lower bounds and settles two small cases, but the main asymptotic question remains open.

The problem asks for the asymptotic minimum number of positive cubature nodes on $S^2$ . The conjectured asymptotic limit has not been established.

September 2026 lower-bound advance

Zhuo Cheng and Deyu Yu claim $N_{2m+1}\ge (m+1)(m+2)+2$ for $m\ge 3$ , establish $N_7=22$ and $N_9=32$ , and improve the asymptotic construction bound. These results narrow the gap but do not settle the asymptotic limit; the claims are unverified here.

Current status (as of October 2026): The cited preprint claims stronger lower bounds and exact values for $N_7$ and $N_9$ , while the asymptotic limit conjecture remains open.

Sources

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