Lattice points in bodies of revolution in \({\mathbb R}^3\): An \(\Omega \_\)-estimate for the error term (Q1977803)

Let \(A_{\mathcal B}(t)\) denote the number of triples in \(\mathbb Z^3\) in the ``blown up'' region \(\sqrt{t}\mathcal B\), where \(\mathcal B\) is a compact, convex body of revolution in \(\mathbb R^3\) which contains the origin as an inner point and satisfies some additional (reasonable) conditions. The author proves that, as \(t\to+\infty\), \[ A_{\mathcal B}(t) = \text{ vol} ({\mathcal B}) t^{3/2} + \Omega_-\left(t^{1/2}(\log t)^{1/3}(\log\log t)^{\log 2\over 3} \text{ e}^{-C\sqrt{\log\log\log t}}\right) \] for some absolute, positive constant \(C\). This is achieved by combining the approaches of \textit{W. G. Nowak} [Arch. Math. 47, 232-237 (1986; Zbl 0594.10042)] and \textit{J. L. Hafner} [Invent. Math. 63, 181-186 (1981; Zbl 0458.10031)]. A crucial point in the proof is to choose the axis of revolution to be one of the coordinate axes. The above result is slightly sharper than the general result of W. G. Nowak (op. cit.), who had \(\Omega_-\left(t^{1/2}(\log t)^{1/3}\right)\).