The entropy of a half-BPS black hole in $N=4$ supersymmetric heterotic string compactification is independent of the details of the charge vector and is a function only of the norm of the charge vector calculated using the appropriate Lorentzian metric. Thus, in order for this to agree with the degeneracy of the elementary string states, the latter must also be a function of the same invariant norm. We show that this is true for generic CHL compactifications to all orders in a power series expansion in the inverse charges, but there are exponentially suppressed corrections which do depend on the details of the charge vector. This is consistent with the hypothesis that the black hole entropy reproduces the degeneracy of elementary string states to all orders in a power series expansion in the inverse charges, and helps us extend the earlier conjectured relation between black hole entropy and degeneracy of elementary string states to generic half-BPS electrically charged states in generic $N=4$ supersymmetric heterotic string compactification. Using this result, we can also relate the black hole entropy to the statistical entropy calculated using an ensemble of elementary string states that contains all BPS states along a fixed null line in the lattice of electric charges.
"Black holes and the spectrum of half-BPS states in $N=4$ supersymmetric string theory." Adv. Theor. Math. Phys. 9 (4) 527 - 558, August 2005.