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Statement

Notation (p. 125): $f$ is a monic polynomial (1), $E$ the set where $|f|<1$ and $\bar E$ its closure.

Problem 10 (p. 142). "If $f$ is of the form (1), does there exist a straight line $\lambda$ such that the projection of $\bar E$ on $\lambda$ has measure at most 2? If $z_0$ is a point of $\bar E$ lying on a line of support of $E$, does $\bar E$ contain a point $z$ such that $|z-z_0|\geq2$?"

Problem 15 (p. 143) refers back to this problem.

Source. P. Erdős, F. Herzog, G. Piranian, Metric properties of polynomials, J. Analyse Math. 6 (1958), 125--148, doi:10.1007/BF02790232; Problem 10 on p. 142. The copy read is identified on the source card.

Read depth. Claims checked: the problem was read clause by clause on the page image of p. 142 on 2026-10-08. Nothing here is independently reviewed.

Dependencies

None.

Bears on

• #1043: the problem is the first question of Problem 10, stated for $\{|f|\le1\}$. That set is $\bar E$: $|f|$ has no local minimum where it equals $1$, so each point with $|f|=1$ is a limit of points with $|f|<1$ (an observation made here). The second question is not part of the problem.