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 tpt, t 2 q : 0 ď t ď 1u to studying l 2 L p{3 decoupling for the projection of this subset to the interval r0, 1s. 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 r0, 1s. Our ideas are inspired by the recent work on ellipsephic sets by Biggs [1, 2] using nested efficient congruencing.

A decoupling interpretation of an old argument for Vinogradov's Mean Value Theorem

“…The study of this dictionary has led to new proofs of Fourier decoupling for the parabola [23], cubic moment curve [15], and the degree k moment curve [16], inspired from the efficient congruencing arguments in [26, Section 4], [19], and [36], respectively. Additionally, a decoupling interpretation of the study of VMVT over ellipspephic sets [1] led to a proof of Fourier decoupling for fractal sets on the parabola [5].…”