A first-moment improvement for unshifted rank-one lattice rules in unanchored Sobolev spaces

Abstract

We study deterministic unshifted rank-one lattice rules in weighted unanchored Sobolev spaces of smoothness one. Let ${\overline{e}}^2\left(n\right)$ denote the average, over all non-zero generating vectors, of the squared worst-case error of an $n$-point rule. For odd prime $n$, the averaged Kazashi–Sloan–Goda inequality reduces the analysis to integer moments of an arithmetic quantity $T_n\left(\kappa \right)+{\delta }_n$. The finite-field computation giving the first moment of $T_n$ is implicit in earlier work; the new step here is to retain that first moment in exact ${\ell }^1$ form and combine it with the uniform cap $T_n\left(\kappa \right)\le {\pi }^2/6$. This gives an ${\ell }^1$-${\ell }^{\infty }$ conversion controlling all integer moments required by the averaged inequality. Consequently, for every odd prime $n$ and finite dimension $d$, ${\overline{e}}^2\left(n\right)$ is bounded by $n^{-1}{(logn)}^2$ times an explicit weighted subset sum. For product weights ${\gamma }_u={\prod }_{j\in u}{\gamma }_j$ satisfying ${\sum }_{j\ge 1}{\gamma }_j<\infty$, this subset sum is bounded independently of $d$. Hence there exists an unshifted rank-one lattice rule with worst-case error $O_{\gamma }(n^{-1/2}logn)$, uniformly in the dimension. Numerical computations illustrate the first-moment identity, the scaled moment conversion, and the dimension dependence of the bounds.