  We prove a strong relation between Chern and log Chern invariants of
algebraic surfaces. For a given arrangement of curves, we find nonsingular
projective surfaces with Chern ratio arbitrarily close to the log Chern ratio
of the log surface defined by the arrangement. Our method is based on sequences
of random p-th root covers, which exploit a certain large scale behavior of
Dedekind sums and lengths of continued fractions. We show that randomness is
necessary for our asymptotic result, providing another instance of "randomness
implies optimal". As an application over the complex numbers, we construct
nonsingular simply connected projective surfaces of general type with large
Chern ratio. In particular, we improve the Persson-Peters-Xiao record for Chern
ratios of such surfaces.
