Decoupling for fractal subsets of the parabola

Abstract

We consider decoupling for a fractal subset of the parabola. We reduce studying l(2)L(p) dccoupling for a fractal subset on the parabola {(t , t(2)) : 0 <= t <= 1} to studying l(2)L(p/3) decoupling for the projection of this subset to the interval [0, 1]. This generalizes the decoupling theorem of Bourgain-Demeter in the case of the parabola. Due to the sparsity and fractal like structure, this allows us to improve upon Bourgain-Demeter's decoupling theorem for the parabola. In the case when p/3 is an even integer we derive theoretical and computational tools to explicitly compute the associated decoupling constant for this projection to [0, 1]. Our ideas are inspired by the recent work on ellipsephic sets by Biggs (arXiv:1912.04351, 2019 and Acta Arith. 200(4):331-348, 2021) using nested efficient congruencing.