Decoupling for fractal subsets of the parabola

Abstract

We consider decoupling for a fractal subset of the parabola. We reduce studying \(l^{2}L^{p}\) decoupling for a fractal subset on the parabola \(\{(t, t^2) : 0 \le t \le 1\}\) 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.