On Markoff's inequality (Q1862751)

The author shows that there are sets of measure zero with Markoff factors of order \(n^2\). In addition, he shows that any set \(E\subset [0,1]\) having a Markoff factor of order \(n^2\) must have a positive logarithmic capacity. Furthermore, he shows a conjecture of Kroo and Szabados is false by constructing a set \(E\subset \mathbb{R}^2\) and a boundary point \(S\in\partial E\) such that \(S\) can be reached from the interior of \(E\) by a \(C^\infty\)-curve with \(E\) having a local Markoff factor \(M_n(S)\) exceeding \(e^n\). With respect to this construction, he shows that local Markoff factors at \(S\) grow exponentially if \(S\) can be reached by an analytic curve.