---

P. Erdős, F. Herzog and G. Piranian, Metric properties of polynomials, J. Analyse Math. 6 (1958), 125-148, pose the problem as their Problem 4 (p. 135) and note (pp. 135-136) that for a line segment or a circular disc the vanishing follows from Chebyshev polynomials with their Theorem 8, and from their Theorem 4, respectively: such an $F$ of transfinite diameter at least $1$ has $\mu(F)=0$. The paper's [[../library/polynomials/erdos_1958_metric_properties_polynomials/_index|library card]] records its theorems. Ghosh and Ramachandran's Theorem 4.1, on their page, gives a rate for the segment $[-2,2]$.

Covers. The second question for a line segment of length at least $4$ or a closed disc of radius at least $1$; nothing about other sets or about the first question.

Refereed. J. Analyse Math. 6 (1958), 125-148. The site's commentary credits the result on a problem it labels OPEN, which is not acceptance.

Depends on. Nothing in this wiki.