Infinite sequences with optimal diaphony, periodic $L_2$-discrepancy, and beyond

Abstract

We investigate the periodic $L_2$-discrepancy of infinite sequences $\hat{A}{\S }_d$ in $[0,1{)}^d$ and its analytic counterpart, the diaphony. We prove that infinite order-2 digital sequences over ${\mathbb{F}}_2$ attain the optimal order $L_{2,N}^{\mathrm{per}}(\hat{A}{\S }_d)\le C_d(\log N{)}^{d/2}/N$ for all $N\in \mathbb{N}\setminus \{1\}$, matching known lower bounds for infinitely many $N\in \mathbb{N}$. This confirms the conjectured optimality of order-2 constructions. By this result, we improve upon previously known constructions using order-5 digital sequences, and reduce the underlying dimension for the interlacing construction from $5d$ to $2d$, significantly improving practicality. We establish our bounds within a broader framework of quasi-Monte Carlo integration for periodic Besov spaces $S_{p,q}^{r}B({\mathbb{T}}^d)$ with dominating mixed smoothness $r\in (1/p,2)$, where $p,q\in [1,\infty ]$. Rules based on infinite order-2 digital sequences yield worst-case errors of order $(\log N{)}^{(d-1)(1-1/q)}/N^{\min (r,1)}$ for $r\ne 1$, and $(\log N{)}^{d(1-1/q)}/N$ for $r=1$, for all $N\in \mathbb{N}\setminus \{1\}$, while preserving extensibility in $N$.