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 Bretèche for the study of the set ${\mathcal{D}}_n$. In particular, using these ideas, we establish that $$\mathrm{\mid }\{(d_1,d_2,d_3)\in {\mathcal{D}}_n^3:d_1+d_2=d_3\}\mathrm{\mid }\le \tau (n{)}^{2-\delta }$$ with $\delta =0.045072$.