On the lattice rest of a convex body in \(\mathbb{R}^s\) (Q800966)

Let \(\mathcal B\) be a compact convex subset of \(\mathbb{R}^s\) \((s\geq 2)\) containing the origin as an inner point and suppose that the boundary \(\partial\mathcal B\) is of class \(C^{\infty}\) and has finite non-vanishing Gaussian curvature throughout. Denote by \(A(t)\) the number of lattice points (of \(\mathbb{Z}^s)\) in the ``blown up'' body \(\sqrt{t}\mathcal B\) and by \(P(t)\) the ``lattice rest'' \(P(t)=A(t) - Vt^{s/2}\) \((V\) the volume of \(\mathcal B)\). Then it had been proved by \textit{E. Hlawka} [Monatsh. Math. 54, 1--36 (1950; Zbl 0036.30902)] that \(P(t)=O(t^{s(s-1)/2(s+1)})\) and \(P(t)=\Omega (t^{(s-1)/4})\). The purpose of the present note is to improve this lower bound to \[ P(t)=\Omega (t^{(s-1)/4}(\log t)^{1/4}). \] The argument makes use of a mean value principle of \textit{M. Riesz} [Acta Sci. Szeged 1, 114--126 (1923), a convolution method of \textit{J. L. Hafner} [Invent. Math. 63, 181--186 (1981; Zbl 0458.10031)] and asymptotic formulas for exponential integrals over such general bodies \(\mathcal B\) due to E. Hlawka (loc. cit.).