On Markoff's Inequality

For compact subsets of the real line the Markoff factors increase at least as fast as n ² . In this paper it is shown that there are sets of measure zero with Markoff factors of order n ² . We shall also show that this cannot happen for compact sets of logarithmic capacity zero. In connection with Markoff's inequality for polynomials of two variables we show a set E R^2 and a boundary point S that can be reached from the interior of E by a C^{∈fin} curve and still the local Markoff factor at S increases exponentially. This solves a conjecture of Kroó and Szabados. We shall also show that this cannot happen if S can bereached by an analytic curve.