Small cap decouplings

Abstract

We develop a toolbox for proving decouplings into boxes with diameter smaller than the canonical scale. As an application of this new technique, we solve three problems for which earlier methods have failed. We start by verifying the small cap decoupling for the parabola. Then we find sharp estimates for exponential sums with small frequency separation on the moment curve in R 3 . This part of the work relies on recent improved Kakeya-type estimates for planar tubes, as well as on new multilinear incidence bounds for plates and planks.We also combine our method with the recent advance on the reverse square function estimate, in order to prove small cap decoupling into square-like caps for the two dimensional cone.The Appendix by Roger Heath-Brown contains an application of the new exponential sum estimates for the moment curve, to the Riemann zeta-function.

A sharp square function estimate for the cone in $\mathbb{R}^3$

“…This problem was raised by Bourgain and Watt [5] in their work on the Gauss circle problem. The paper [9] shows that the square function estimate Theorem 1.1 implies a sharp estimate for this decoupling problem.…”

“…When f θ L p (BR) is close to its largest value, then |f θ | is concentrated on a sparse region of B R . The argument in [9] exploits this sparsity to improve the bound from decoupling and give sharp estimates for f L p (BR) for every p. The proof of the main theorem here builds on that proof.…”

“…Theorem 1.3 makes this precise. There are similar issues in the problem of decoupling into small caps, which was studied in [9]. For instance, consider an exponential sum of the form…”

“…The paper [9] also considers a decoupling problem in which the cone is divided into small squares instead of sectors. This problem was raised by Bourgain and Watt [5] in their work on the Gauss circle problem.…”

“…Strictly speaking, one need to apply Lemma 6.1 and Lemma 6.2 to justify the first " " in inequality(9). This is similar to the arguments in Section 6 where we do in full details.…”