Existence of good lattice points in the sense of Hlawka (Q1252892)

Good lattice points modulo an integer \(m\ge 2\) are of importance in a numerical integration technique for periodic functions of several variables developed by \textit{E. Hlawka} [Monatsh. Math. 66, 140--151 (1962; Zbl 0105.04603)] and \textit{N. M. Korobov} [Dokl. Akad. Nauk SSSR 124, 1207--1210 (1959; Zbl 0089.04201)] and in the work of the author on the multidimensional distribution behaviour of linear congruential pseudo-random numbers [Adv. Math. 26, 99--181 (1977; Zbl 0366.65004); Bull. Am. Math. Soc. 84, 957--1041 (1978; Zbl 0404.65003)]. The existence of good lattice points was so far only known for prime moduli (N. M. Korobov [loc. cit.]). In the present paper, an existence theorem for good lattice points is established for general moduli by using a method based on exponential sums. The detailed result reads as follows. For a nonzero lattice point \(\mathfrak h = (h_1, \ldots, h_s) \in \mathbb Z^s\), \(s\ge 2\), let \(r(\mathfrak h)\) be the absolute value of the product of the nonzero coordinates of \(\mathfrak h\). For \(\mathfrak g\in \mathbb Z^2\) define \(R(\mathfrak g,m)\) to be the sum of the numbers \(r(\mathfrak h)^{-1}\) extended over those nonzero lattice points \(\mathfrak h\) with \(-m/2<h_j\le m/2\) for \(1\le j\le s\) and \(\mathfrak h\cdot \mathfrak g \equiv 0 \pmod m\). Then for every dimension \(s\ge 2\) there exists a lattice point \(\mathfrak g\in\mathbb Z^s\) with coordinates relatively prime to \(m\) and \[ R(\mathfrak g,m) < m^{-1} \left(\frac75 + 2\log m\right)^s.\] A slightly improved result can be shown in case \(m\) is a prime power.