Erdős Problem #1151 — Prescribed limit sets at a Chebyshev interpolation point

Erdős Problem $1151$ asks whether every closed subset of $[-1,1]$ can be realized as the finite cluster set of interpolation values at a prescribed evaluation point. Erdős claimed this in $1943$ for certain points, but the claim was published without proof; the formulation may be ambiguous about whether the point is fixed arbitrarily.

Known results

• Erdős, $1941$: for $x=\cos(\pi p/q)$ with odd $p,q\geq 1$, some continuous function has interpolation values diverging to infinity.
• Erdős, $1943$: claimed, without proof, that every closed set can occur at such points.
• Marcinkiewicz and Grünwald, $1936$: some continuous function has Chebyshev interpolants diverging at every point of $[-1,1]$; this does not prescribe finite cluster sets.

Formalization attempt

A formal-conjectures entry records missing definitions and formalization components, but supplies no proof, counterexample, or named AI contribution.

Current status (as of March 2026): Erdős’s divergence result and unproved arbitrary-closed-set claim are known, but the prescribed finite-cluster-set problem remains unresolved.