An $l^2$ decoupling interpretation of efficient congruencing: the parabola

Abstract

We give a new proof of $l^2$ decoupling for the parabola inspired from efficient congruencing. Making quantitative this proof matches a bound obtained by Bourgain for the discrete restriction problem for the parabola. We illustrate similarities and differences between this new proof and efficient congruencing and the proof of decoupling by Bourgain and Demeter. We also show where tools from decoupling such as $l^2{L}^2$ decoupling, Bernstein’s inequality, and ball inflation come into play.