Higher moments for lattice point discrepancy of convex domains and annuli.

• Integer lattice. • Discrepancy. • Annuli. • Gauss circle problem. Given a domain Ω ⊆ R 2 , let D (Ω , x , R) be the number of lattice points from Z 2 in R Ω − x , for R ≥ 1 and x : = (x 1 , x 2) ∈ T 2 , minus the area of R Ω: D (Ω , x , R) = # { (j , k) ∈ Z 2 : (j − x 1 , k − x 2) ∈ R Ω } − R 2 | Ω |. We call ∫ T 2 | D (Ω , x , R) | p d x the p -th moment of the discrepancy function D. In 2014, Huxley showed that for convex domains with sufficiently smooth boundary, the fourth moment of D is bounded by O (R 2 log ⁡ R) , and in 2019, Colzani, Gariboldi, and Gigante extended this result to higher dimensions. In this paper, our contribution is twofold: first, we present a simple direct proof of Huxley's 2014 result; second, we establish new estimates for the p -th moments of lattice point discrepancy of annuli of radius R , and any fixed thickness 0 < t < 1 for p ≥ 2. For a video summary of this paper, please visit https://youtu.be/YWIe1IBIi9Q. [ABSTRACT FROM AUTHOR]