On the lattice point problem for ellipsoids
Let \(Q(x)\) be a positive definite quadratic form for \(x\in \mathbb{R}^k\) and let \(E_s = \{x\in \mathbb{R}^k: Q(x) \leq s\}\) denote an ellipsoid of dimension \(k\) with a large parameter \(s\). Let \(A_s\) be the number of lattice points in the ellipsoid \(E_s\). As usual, the lattice rest is defined by \(P_s= A_s- Vs^{k/2}\), where \(V\) is the volume of \(E_1\). For the general class of ellipsoids, E. Landau found for \(P_s\) the estimation \(P_s\ll s^{{k \over 2}-1+{1 \over k+1}}\), which was extended by E. Hlawka to convex bodies with smooth boundary and strictly positive Gaussian curvature. This result was improved by \textit{W. G. Nowak} and \textit{E. Krätzel} [Acta Arith. 62, 285-295 (1992; Zbl 0769.11037)] to the order \(s^{{k\over 2}-1+ \lambda +\varepsilon}\) with \(\lambda= 4/(5k+2)\) for \(3\leq k\leq 6\) and \(\lambda= 3/4k\) for \(k \geq 7\), \(\varepsilon >0\). For special ellipsoids some improvements were obtained by Diviš, Jarnḱ and Novák. For example, \[ P_s\ll s^{{k \over 2}-1} \] holds for rational and diagonal quadratic forms with \(k\geq 5\). The authors now prove the same estimation for all ellipsoids with \(k\geq 9\) by means of probabilistic methods. Only a short sketch of the proof is given.
scientific article; zbMATH DE number 1050583
scientific article