A Continuum of Small-cap Decouplings and Exponential Sums for the Moment Curve in ${\mathbb{R}}^4$

Abstract

We use the high-low method and wavepacket pruning to prove new small-cap decoupling estimates for the moment curve in ${\mathbb{R}}^4$. As an application, we verify a conjecture of Demeter regarding the $L^{12}$ square-root cancellation of exponential sums associated with the moment curve in ${\mathbb{R}}^4$. This provides a continuum of square-root cancellation estimates that connects the Vinogradov MVT in ${\mathbb{R}}^3$ with a result of Bourgain, related to improving the best-known estimate for the Lindelöf hypothesis.

23 pages, 1 figure