Optimal coefficients modulo prime powers in the three-dimensional case
Good lattice points and optimal coefficients were introduced by \textit{E. Hlawka} [Monatsh. Math. 66, 140-151 (1962; Zbl 0105.046)] and \textit{N. M. Korobov} [Dokl. Akad. Nauk SSSR 124, 1207-1210 (1959; Zbl 0089.042)]. Later Niederreiter established links between optimal coefficients and the multidimensional distribution behaviour of linear congruential pseudorandom numbers. For a dimension \(s\geq 2\) and a modulus \(M\geq 2\), an s-dimensional lattice point \(g=(g_ 1,...g_ 2)\in {\mathbb{Z}}^ s\) is called a good lattice point mod M if the ``figure of merit'' \(\rho (g,M)=\min \prod^{s}_{i=1}\max (1,| h_ i|)\) is large (the minimum is extended over all lattice points \((h_ 1,...,h_ s)\neq (0,...,0))\). \textit{S. K. Zaremba} [Monatsh. Math. 78, 446-460 (1974; Zbl 0292.10023)] proved the existence of good lattice points g such that \[ (*)\quad \rho (g,M)\geq \frac{C_ s M}{(\log M)^{s-1}}. \] For the applications to the linear congruential method the important case is M a prime-power and \(g=(1,x,x^ 2,...,x^{s-1})\), x a primitive root mod M. Appropriate existence theorems for such lattice points for primes M and \(s\geq 2\) and for prime powers M and \(s=2\) were shown by \textit{H. Niederreiter} [Adv. Math. 26, 99-181 (1977; Zbl 0366.65004)]. In the present paper the existence of such lattice points \(g=(1,x,x^ 2)\) satisfying (*) in dimension \(s=3\) is proved.
linear congruential pseudorandom numbers
scientific article; zbMATH DE number 4143323