The Concentration Function of Additive Functions with Nonmultiplicative Weight

Suppose that $g(n)$ is a real-valued additive function and $\tau(n)$ is the number of divisors of $n$. In this paper, we prove that there exists a constant $C$ such that
$$\sup_a\sum_{\substack n<N\\g(n)\in[a,a+1)} \tau(N-n) \le C\frac{N\,\log N}{\sqrt{W(N)}},$$
where
$$W(N) =4+\min_\lambda\biggl(\lambda^2 +\sum_{p<N} \frac1p\min\bigl(1,(g(p)-\lambda\log p)^2\bigr)\biggr).$$
In particular, it follows from this result that
$$\sup_a\bigl|\bigl\{m,n:mn<N,\;g(N-mn)=a\bigr\}\bigr| \ll N\,\log N\, \biggl(\sum_{p<N,\,g(p)\ne0}(1/p)\biggr)^{-1/2}.$$
The implicit constant is absolute.