Number of integers represented by families of binary forms (II): binomial forms

We consider some families of binary binomial forms aX(d) + bY(d), with a and b integers. Under suitable assumptions, we prove that every rational integer m with vertical bar m vertical bar >= 2 is only represented by a finite number of forms of this family (with varying d, a, b). Furthermore, the number of such forms of degree >= d(0) representing m is bounded by O(vertical bar m vertical bar(1/d0+epsilon)) uniformly for vertical bar m vertical bar >= 2. We also prove that the integers in the interval [-N, N] represented by one of the forms of the family of degree d >= d(0) are almost all represented by some form of the family of degree d = d(0) if such forms of degree d(0) exist. In a previous paper we investigated the particular case where the binary binomial forms are positive definite. We now treat the general case by using a lower bound for linear forms in logarithms.