A note on unshifted lattice rules for high-dimensional integration in weighted unanchored Sobolev spaces

This short article studies a deterministic quasi-Monte Carlo lattice rule in weighted unanchored Sobolev spaces of smoothness 1 . Building on the error analysis by Kazashi and Sloan, we prove the existence of unshifted rank-1 lattice rules that achieve a worst-case error of O ( n − 1 / 4 ( log n ) 1 / 2 ) , with the implied constant independent of the dimension, under certain summability conditions on the weights. Although this convergence rate is inferior to the one achievable for the shifted-averaged root mean squared worst-case error, the result does not rely on random shifting or transformation and holds unconditionally without any conjecture, in contrast to the result of Kazashi and Sloan.