Asymptotic degeneracy of dyonic N=4 string states and black hole entropy

It is shown that the asymptotic growth of the microscopic degeneracy of BPS dyons in four-dimensional N = 4 string theory captures the known corrections to the macroscopic entropy of four-dimensional extremal black holes. These corrections are subleading in the limit of large charges and originate both from the presence of interactions in the effective action quadratic in the Riemann tensor and from non-holomorphic terms. The presence of the non-holomorphic corrections and their contribution to the thermodynamic free energy is discussed. It is pointed out that the expression for the macroscopic entropy, written as a function of the dilaton field, is stationary at the horizon by virtue of the attractor equations.