Distribution of Korobov-Hlawka sequences

Abstract

Let $a_1, …, a_s$ be integers and $N$ be a positive integer. Korobov (1959) and Hlawka (1962) proposed to use the points $$ x^{(k)}=(\{\frac{a_1 k}N\}, …, \{\frac{a_1 k}N\}), \qquad k=1,…, N, $$ as nodes of multidimensional quadrature formulae. We obtain some new results related to the distribution of the sequence $K_N(a)=\{x^{(1)},…,x^{(N)}\}$. In particular, we prove that $$ \frac{\ln^{s-1} N}{N \ln\ln N} \underset{s}\ll D(K_N(a)) \underset{s}\ll \frac{\ln^{s-1} N}{N} \ln\ln N $$ for ‘almost all’ $a\in (\mathbb Z_N^*)^s$, where $D(K_N(a))$ is the discrepancy of the sequence $K_N(a)$ from the uniform distribution and $\mathbb Z^*_N$ is the reduced system of residues modulo $N$. Bibliography: 18 titles.