A mean-square bound concerning the lattice discrepancy of a torus in \(\mathbb R^3\)
Let \(\mathcal{B}\) denote a compact body in three-dimensional Euclidean space, and \(t\) a large real parameter. The lattice point discrepancy of a linearly dilated copy \(t\mathcal{B}\) is defined by \(P_{\mathcal{B}}(t)=\#(t\mathcal{B} \cap\mathbb{Z}^3)-\text{vol}(\mathcal{B})t^3\). Let \(\mathcal{T}=\mathcal{T}_{a,b}\) be a torus bounded by \[ \partial \mathcal{T}_{a,b}: \begin{pmatrix} x\\ y \\z \end{pmatrix} =\begin{pmatrix} (a+b\cos \alpha)\cos \beta \\ (a+b\cos \alpha)\sin \beta\\ b\sin \alpha \end{pmatrix},\quad 0\leq \alpha, \beta <2\pi, \] where \(a>b>0\) are fixed constants. \textit{W. G. Nowak} [Acta Math. Hung. 120, 179--192 (2008; Zbl 1174.11079)] proved that for \(t\) large \(P_{\mathcal{T}}(t)=\mathcal{F}_{a,b}(t)t^{3/2}+\Delta_{\mathcal{T}}(t)\), where \(\mathcal{F}_{a,b}(t)=4a\sqrt{b}\sum_{j=1}^{\infty}j^{-3/2}\sin(2\pi jbt-\pi/4)\) and \(\Delta_{\mathcal{T}}(t)\ll t^{11/8+\varepsilon}\). In this paper the authors show that \[ \int_0^T(\Delta_{\mathcal{T}} (t))^2=O(T^{3+\varepsilon}) \] for each \(\varepsilon>0.\)
scientific article; zbMATH DE number 5881198