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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 numberof lattice points in thin annuli. As a consequence, we enhance thewell-posedness level of the periodic quintic Nonlinear Schr\"odinger equationto $s > \frac{131}{624}\sim 0.21$, which is an improvement on the results of DeSilva, Pavlovi\'c, Staffilani, & Tzirakis, Li, Wu, & Xu, and Schippa. We alsopresent conditional results, dependent on improvements on the count of latticepoints in thin annuli.

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