An application of Diophantine approximation to the construction of rank-1 lattice quadrature rules

Lattice quadrature rules were introduced by Frolov (1977), Sloan (1985) and Sloan and Kachoyan (1987). They are quasi-Monte Carlo rules for the approximation of integrals over the unit cube in ${\mathbb {R}}^{s}$ and are generalizations of `number-theoretic' rules introduced by Korobov (1959) and Hlawka (1962)---themselves generalizations, in a sense, of rectangle rules for approximating one-dimensional integrals, and trapezoidal rules for periodic integrands. Error bounds for rank-1 rules are known for a variety of classes of integrands. For periodic integrands with unit period in each variable, these bounds are conveniently characterized by the figure of merit $\rho$, which was originally introduced in the context of number-theoretic rules. The problem of finding good rules of order $N$ (that is, having $N$ nodes) then becomes that of finding rules with large values of $\rho$. This paper presents a new approach, based on the theory of simultaneous Diophantine approximation, which uses a generalized continued fraction algorithm to construct rank-1 rules of high order.