For fixed real numbers \(k>2\), \(n\geq 3\), and large \(x\), let \(R_{k,n} (x)\) denote the lattice point number defined in the title. According to the first named author’s monograph [Lattice points (1988; Zbl 0675.10031)], one has an asymptotic representation \[ R_{k,n} (x)= V_{k,n} x^{n/k}+ \sum_{r=1}^{n-1} H_{k,n, r} (x)+ \Delta_{k,n} (x), \tag{1} \] where \(V_{k,n} x^{n/k}\) is the volume of the domain in question, \(\Delta_{k,n} (x)\) is an error term and the functions \(H_{k,n,r} (x)\) are defined in terms of integral representations involving \(\Delta_{k,r} (x)\). In order to get sharp estimates for the remainder terms in (1), it is of interest to have at hand precise bounds for \(\int_ 0^ x \Delta_{k,r} (t) dt\). In this direction, the main achievement of the present paper is the result \[ \int_ 0^ x \Delta_{k,3} (t) dt\ll x\log x. \tag{2} \] This is established by most tricky arguments involving the theory of two- dimensional exponential sums. Combining (2) with a corresponding known bound for the case \(r=2\), the authors obtain significantly improved estimates for the functions \(H_{k,n,r} (x)\).
For a part, these are capable of further slight refinements, if one incorporates the results of M. N. Huxley’s recent paper [Proc. Lond. Math. Soc., III. Ser. 66, 279-301 (1993; Zbl 0820.11060)].