Problem 988

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claims/: The 1 claim page of Problem 988, one per claimant's result; the problem's standing derives from them.

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Statement. If $P\subseteq S^2$ is a subset of the unit sphere then define the discrepancy

$$D(P) = \max_C \lvert \lvert C\cap P\rvert - \alpha_C \lvert P\rvert \rvert,$$

where the maximum is taken over all spherical caps $C$, and $\alpha_C$ is the appropriately normalised measure of $C$.

Is it true that

$$\min_{\lvert P\rvert=n}D(P)\to \infty$$

as $n\to \infty$?

Status. SOLVED, the site's label: Schmidt's 1969 theorem, refereed in Inventiones Mathematicae, answers the question yes in every dimension. The derived standing, solved and proved, is more specific than the site's label, which names no polarity: the accepted claim on the claim page proves the statement.

Source. erdosproblems.com/988, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #988, https://www.erdosproblems.com/988.

References.

• [Ro54] Roth, K. F., On irregularities of distribution. Mathematika (1954), 73-79.
• [Sc69b] Schmidt, Wolfgang M., Irregularities of distribution. IV. Invent. Math. (1969), 55-82.

Formalization. No formal-conjectures statement: the site's indicator reports no formalized statement and the community database records none. The file src/latest/ErdosProblems/Erdos988.lean of the GitHub repository plby/lean-proofs declares itself a Lean formalization of a solution to the problem, with Schmidt as its informal author and Codex and GPT-5.6 Sol as its formal authors; it proves $\lvert P\rvert\le512\,D(P)^4$ for every finite $P\subseteq S^2$ and from it the divergence of the minimum discrepancy, by a Stolarsky positive-kernel argument rather than Schmidt's. It is linked, pinned, from Schmidt's claim page, and is not built or audited here, so no claim carries formalized evidence.

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