The principle of simultaneous saturation: Application to the $k$-linear restriction/extension problem

This paper develops a new framework, \emph{simultaneous saturation}, designed to quantify the size of sets whose elements are simultaneously large. The framework establishes a correspondence between the magnitude of such sets and a system of interdependent conditions linking their points. We first prove a general theorem establishing the correspondence and then apply the framework to multilinear restriction-type estimates. From this perspective, we obtain a new proof (independent of Bennett-Carbery-Tao \cite{BCT}) of the $d$-linear restriction/extension theorem, and establish the ${\lambda }^{\epsilon }$ loss conjectured bounds for the $k$-linear $L^2\to L^{p\mathrm{/}k}$ extension problem under mixed transversality/curvature conditions $(k<d)$.