On lattice points, short-time estimates, and global well-posedness of the quintic NLS on <inline-formula><tex-math id="M1">$ \mathbb{T} $</tex-math></inline-formula>

We prove and utilize an improvement to the short time estimates of Burq, Gérard, & Tzvetkov [7] on $ \mathbb{T} $ via connecting this estimate to the number of lattice points in thin annuli. As a consequence, we enhance the global 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, [12], Li, Wu, & Xu, [26], and Schippa [27]. We also present conditional results, dependent on improvements on the count of lattice points in thin annuli.