Decoupling for fractal subsets of the parabola

TL;DR

This paper extends decoupling theory to fractal subsets of the parabola, reducing the problem to a simpler case and improving existing bounds, with explicit computations for certain parameters.

Contribution

It generalizes Bourgain-Demeter's decoupling theorem to fractal parabola subsets and introduces methods for explicit decoupling constant calculations.

Findings

Reduced decoupling problem to interval projection

Improved bounds for fractal parabola subsets

Explicit computation methods for specific p/3 values

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 $\{(t, t^2) : 0 \leq t \leq 1\}$ 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.