On Lattice Points, Short-Time Estimates, and Global Well-posedness of the Quintic NLS on $\mathbb{T}$

TL;DR

This paper improves short-time estimates for the periodic quintic NLS on the torus by linking lattice point counts in thin annuli to well-posedness, achieving a lower regularity threshold than previous results.

Contribution

It introduces a novel connection between lattice point counting and short-time estimates, leading to enhanced well-posedness results for the quintic NLS on the torus.

Findings

Enhanced well-posedness threshold to s > 0.21

Improved short-time estimates via lattice point analysis

Conditional results based on lattice point count improvements

Abstract

We prove and utilize an improvement to the short time estimates of Burq, G\'erard, & Tzvetkov on $\mathbb{T}$ via connecting this estimate to the number of lattice points in thin annuli. As a consequence, we enhance the well-posedness level of the periodic quintic Nonlinear Schr\"odinger equation to $s > \frac{131}{624}\sim 0.21$, which is an improvement on the results of De Silva, Pavlovi\'c, Staffilani, & Tzirakis, Li, Wu, & Xu, and Schippa. We also present conditional results, dependent on improvements on the count of lattice points in thin annuli.