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 using nested efficient congruencing.

28 pages, typos corrected, references updated, to appear in Math. Z