On the exceptional set for binary Egyptian fractions

For fixed integer a ≥ 3, we study the binary Diophantine equation a/n = 1/x + 1/y and in particular the number Ea(N) of n ≤ N for which the equation has no positive integer solutions in x and y. The asymptotic formula Ea(N) ∼ C(α)N(log logN)^2m-1-1/(logN)^1-1/2m, as N goes to infinity, is established in this article, and this improves the best result in the literature dramatically. The proof depends on a very delicate analysis of a certain combinatorial property of the underlying group (Z/aZ)* and m depends in a subtle way on the factorization of α.