  In this paper, we provide estimates for the additive discretized energy of
\[\sum_{c\in C} |\{(a_1, a_2, b_1, b_2)\in A^2\times B^2: |(a_1 +cb_1) - (a_2 +
cb_2)|\le \delta\}|_{\delta},\] that depend on non-concentration conditions of
the sets. Our proof follows the Guth-Katz-Zahl approach (2021) with appropriate
changes along the way clarifying and optimizing many of the steps. Several
applications will also be discussed.
