A problem of McMillan on conformal mappings. (Q5930561)

Let \(f\) be a conformal mapping of the unit disk \(\mathbb{D}\) onto a simply-connected domain \(\Omega\), and let \(A\) denote the set of all ideal accessible boundary points \(f(e^{i\theta})\) of \(\Omega\) when \(f\) has the non-tangential limit \(f(e^{i\theta})\) at \(e^{i\theta}\). Let \(D(a,r)\) denote the disk with center \(a\) and radius \(r\). For \(a\in A\) and small \(r\), let \(\gamma(a,r)\subset\partial D(a,r)\) be the crosscut of \(\Omega\) separating \(a\) from \(f(0)\) which can be joined to \(a\) by a Jordan arc in \(\Omega\cap D(a,r)\). Let \(L(a,r)\) denote the Euclidean length of \(\gamma(a,r)\) and let \(A(a,r)\) denote the Lebesgue measure of \(U(a,r)=\bigcup_{r'<r}\gamma(a,r')\). \textit{J. E. McMillan} proved in [Duke Math. J. 37, 725--739 (1970; Zbl 0222.30007)], that the set of \(a\in A\) such that NEWLINE\[NEWLINE \limsup_{r\rightarrow 0} \frac{A(a,r)}{\pi r^2} < \frac{1}{2} NEWLINE\]NEWLINE has harmonic measure zero. He also conjectured that the set \(E_1\) of \(a\in A\) for which NEWLINE\[NEWLINE \liminf_{r\rightarrow 0} \frac{A(a,r)}{\pi r^2} > \frac{1}{2} NEWLINE\]NEWLINE and the \(E_2\) for which NEWLINE\[NEWLINE \liminf_{r\rightarrow 0} \frac{L(a,r)}{2\pi r} > \frac{1}{2} NEWLINE\]NEWLINE have both harmonic measure zero. In this paper, the authors establish McMillan's conjecture regarding the set \(E_2\).