Normal numbers and nested perfect necklaces

TL;DR

This paper characterizes Levin's low discrepancy sequence construction using nested perfect necklaces, providing explicit methods and counts for their formation, and extends the discrepancy results to a broader class of real numbers.

Contribution

It introduces a characterization of Levin's sequences via nested perfect necklaces and generalizes the discrepancy bounds to a wider class of real numbers.

Findings

Levin's sequences are characterized by nested perfect necklaces.

Real numbers with base b expansions as concatenations of nested perfect necklaces have low discrepancy.

Explicit construction methods for nested perfect necklaces are provided for base 2.

Abstract

M. B. Levin used Sobol-Faure low discrepancy sequences with Pascal matrices modulo $2$ to construct, for each integer $b$, a real number $x$ such that the first $N$ terms of the sequence $(b^n x \mod 1)_{n\geq 1}$ have discrepancy $O((\log N)^2/N)$. This is the lowest discrepancy known for this kind of sequences. In this note we characterize Levin's construction in terms of nested perfect necklaces, which are a variant of the classical de Bruijn necklaces. Moreover, we show that every real number $x$ whose base $b$ expansion is the concatenation of nested perfect necklaces of exponentially increasing order satisfies that the first $N$ terms of $(b^n x \mod 1)_{n\geq 1}$ have discrepancy $O((\log N)^2/N)$. For base $2$ and the order being a power of $2$, we give the exact number of nested perfect necklaces and an explicit method based on matrices to construct each of them.