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Statement

Notation (p. 125): $f$ is a monic polynomial (1), $E$ the set where $|f|<1$, and $C$ the unit circle; $|E\cap C|$ is one-dimensional measure.

Theorem 5 (p. 134). "For a polynomial (1) with all $z_\nu$ on $C$, the relation $0<|E\cap C|<2\pi$ holds, and the constants 0 and $2\pi$ are the best possible."

Source. P. Erdős, F. Herzog, G. Piranian, Metric properties of polynomials, J. Analyse Math. 6 (1958), 125--148, doi:10.1007/BF02790232; Theorem 5 on p. 134, its proof on pp. 134--135. The copy read is identified on the source card.

Read depth. Claims checked: the statement was read on the page image of p. 134 on 2026-10-08; the proof was read for structure, not checked. Nothing here is independently reviewed.

Proof pointer

Pages 134--135. The strict bounds: $f$ has a zero on $C$, so $|E\cap C|>0$, and $|E\cap C|=2\pi$ would put the maximum of $|f|$ on $\bar D$ at the origin. Sharpness at $0$: start from $z^n-1$ and move to $z=1$ every zero with $|\arg z_\nu|<\varepsilon$; an explicit ratio estimate shows the moved factor tends to infinity uniformly on $C$ off the arc $|\arg z|<\varepsilon$, so $\limsup_n|E(f_n)\cap C|\le2\varepsilon$. Sharpness at $2\pi$: move the zeros with $0<|\arg z_\nu|<\varepsilon$ instead to the nearer of $e^{\pm i\varepsilon}$, which the paper says works similarly.

Dependencies

None.

Bears on

No problem in the catalog is recorded here as concerning this theorem. It studies the zeros-on-$C$ class of the Corollary to Theorem 4 through the arc length of $E\cap C$ rather than the area of $E$.