Lattice point problems and distribution of values of quadratic forms (Q1971909)

Let \(E\) be a \(d\)-dimensional ellipsoid and let \(A(x;E)\) denote the number of lattice points in \(xE\), where \(x\) is a large positive parameter. Then \[ A(x;E) = \text{vol}(E)x^d + \Delta(x;E). \] E. Landau proved for the error term the estimate \[ \Delta(x;E)\ll x^{d-2+2/(d+1)} \] for all \(d\geq 2\). An improvement of this result for \(d\geq 4\) was given by the reviewer and \textit{W. G. Nowak} [Acta Arith. 62, 285--295 (1992; Zbl 0769.11037)]. For special ellipsoids a lot of estimations is available. Especially, A. Walfisz for \(d\geq 9\) and E. Landau for \(d\geq 5\) proved for ellipsoids with rational quadratic forms the estimation \[ \Delta(x; E)\ll x^{d-2}. \] In a former paper the authors [Acta Arith. 80, 101--125 (1997; Zbl 0871.11069)] proved this result for all ellipsoids. Now, in this paper the authors show that, for ellipsoids with irrational quadratic forms, even the estimation \[ \Delta(x;E)=o(x^{d-2}) \] holds. From this result a conjecture of H. Davenport and D. J. Lewis concerning the gaps between successive values of a positive definite irrational quadratic form is proved. Further, Oppenheim's conjecture for indefinite irrational quadratic forms is derived.