Higher moments for lattice point discrepancy of convex domains and annuli

Given a domain $\Omega \subseteq \mathbb{R}^2$, let $\mathcal{D}(\Omega,x,R)$ be the number of lattice points from $\mathbb{Z}^2$ in $R\Omega-x$, for $R \ge 1$ and $x\in \mathbb{T}^2$, minus area $R\Omega$: $$\mathcal{D}(\Omega,x,R) = \# \{ (j,k) \in \mathbb{Z}^2 :(j-x_1,k-x_2) R\Omega \} - R^2|\Omega|.$$ We call $\int_{\mathbb{T}^2}|\mathcal{D}(\Omega,x,R)|^pdx$ $p$-th moment discrepancy function $\mathcal{D}$. In 2014, Huxley showed that convex domains with sufficiently smooth boundary, fourth $\mathcal{D}$ is bounded by $\mathcal{O}(R^2\log R)$, 2019, Colzani, Gariboldi Gigante extended this result to higher dimensions. paper, our contribution twofold: first, we present simple direct proof Huxley's 2014 result; second, establish new estimates moments point annuli radius $R$, any fixed thickness $0<t<1$ $p\ge 2.$