An elementary proof of a certain theorem on integer points (Q1916572)

The author considers the number \(R(x)\) of lattice points in the domain \(u^{2k}+ v^{2k} \leq x\), where \(k\geq 2\) is an integer. An asymptotic representation for \(R(x)\) is proved anew with an error term of order \(x^{1/3 k}\). For \(k=2\) an improved error term is given. But the results and better estimations of the error term can be found in the textbook of the reviewer [Lattice points, Berlin (1988; Zbl 0675.10031)].