Erdős Problem #1044
Resolved in some way other than a straight proof or disproof — usual for open-ended questions, or for multi-part problems whose parts went different ways. A solution has also been checked by machine, in Lean.
What the community database records, not our verdict. erdosproblems.com no longer shows statuses (why).
The polynomials under consideration: those of the form \(f(z)=\prod_{i=1}^n(z-z_i)\), for some \(n\geq 1\), where \(\lvert z_i\rvert\leq 1\) for all \(i\).
\(\Lambda(f)\), the maximum of the lengths of the boundaries of the connected components of \(\{ z: \lvert f(z)\rvert<1\}\), where the length of a subset of \(\mathbb{C}\) is its \(1\)-dimensional Hausdorff measure.
Attempts
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