Discretized sum-product type problems: Energy variants and Applications

In this paper, we provide estimates for the additive discretized energy of $$\sum _{c\in C}\mathrm{\mid }\{(a_1,a_2,b_1,b_2)\in A^2\times B^2:\mathrm{\mid }(a_1+c{b}_1)-(a_2+c{b}_2)\mathrm{\mid }\le \delta \}{\mathrm{\mid }}_{\delta },$$ 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.