A randomised lattice rule algorithm with pre-determined generating vector and random number of points for Korobov spaces with $0 < α\le 1/2$

In previous work (Kuo, Nuyens, Wilkes, 2023), we showed that a lattice rule with a pre-determined generating vector but random number of points can achieve the near optimal convergence of $O(n^{-α-1/2+ε})$, $ε> 0$, for the worst case expected error, commonly referred to as the randomised error, for numerical integration of high-dimensional functions in the Korobov space with smoothness $α> 1/2$. Compared to the optimal deterministic rate of $O(n^{-α+ε})$, $ε> 0$, such a randomised algorithm is capable of an extra half in the rate of convergence. In this paper, we show that a pre-determined generating vector also exists in the case of $0 < α\le 1/2$. Also here we obtain the near optimal convergence of $O(n^{-α-1/2+ε})$, $ε> 0$; or in more detail, we obtain $O(\sqrt{r} \, n^{-α-1/2+1/(2r)+ε'})$ which holds for any choices of $ε' > 0$ and $r \in \mathbb{N}$ with $r > 1/(2α)$.

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