Regularities of the distribution of abstract van der Corput sequences

Similarly to $\beta$-adic van der Corput sequences, abstract van der Corput sequences can be defined for abstract numeration systems. Under some assumptions, these sequences are low discrepancy sequences. The discrepancy function is computed explicitely, and a characterization of bounded remainder sets of the form $[0,y)$ is provided.

💡 Research Summary

The paper introduces a new class of low‑discrepancy sequences built from abstract numeration systems (ANS), extending the classical van der Corput construction beyond the usual base‑$b$ or $\beta$‑adic settings. An ANS consists of a finite alphabet $\Sigma$, a regular language $L\subseteq\Sigma^{*}$, and the lexicographic order on $L$, which yields a bijection between the natural numbers $\mathbb N$ and the words of $L$. For each $n\in\mathbb N$ the $n$‑th word $w_n$ of $L$ is mapped to a real number in $