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

We prove and utilize an improvement to the short time estimates of Burq, Gérard, & 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 Schrödinger equation to $s>\frac{131}{624}\sim 0.21$, which is an improvement on the results of De Silva, Pavlović, Staffilani, & Tzirakis, Li, Wu, & Xu, and Schippa. We also present conditional results, dependent on improvements on the count of lattice points in thin annuli.