Relations in the Set of Divisors of an Integer $n$

Let $\mathcal{D}_{n} \subset \mathbb{N}$ be the set of the $\tau(n)$ divisors of $n$. We generalize a method developed by Erd\H os, Tenenbaum and de la Bret\`eche for the study of the set $\mathcal{D}_{n}$. In particular, using these ideas, we establish that $$|\{(d_{1},d_{2},d_{3}) \in \mathcal{D}_{n}^3 : d_{1}+d_{2}=d_{3}\}| \le \tau(n)^{2-\delta}$$ with $\delta=0.045072$.