M. B. Levin used Sobol–Faure low discrepancy sequences with Pascal triangle matrices modulo 2 to construct, a real number $x$ such that the first $N$ terms of the sequence ${(2^{n}xmod1)}_{n\ge 1}$ have discrepancy $O({(logN)}^2∕N)$. 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 sequences. Moreover, we show that every real number $x$ whose binary expansion is the concatenation of nested perfect necklaces of exponentially increasing order satisfies that the first $N$ terms of ${(2^{n}xmod1)}_{n\ge 1}$ have discrepancy $O({(logN)}^2∕N)$. For 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. The computation of the $n$th digit of the binary expansion of a real number built from nested perfect necklaces requires $O(logn)$ elementary mathematical operations.

MB Levin使用Sobol–Faure低差异序列和Pascal三角矩阵模2来构造实数 $X$ 这样第一个 $ñ$ 序列项 ${（2^{ñ}X模\mathrm{1个}）}_{ñ\ge \mathrm{1个}}$ 有差异 $Ø（{（日志ñ）}^2∕ñ）$。这是此类序列已知的最低差异。在本说明中，我们用嵌套式完美项链来描述Levin的构造，该项链是经典de Bruijn序列的变体。而且，我们表明每个实数$X$ 其二进位扩展是嵌套的完美项链的指数级递增连接，这满足了第一个 $ñ$ 条件 ${（2^{ñ}X模\mathrm{1个}）}_{ñ\ge \mathrm{1个}}$ 有差异 $Ø（{（日志ñ）}^2∕ñ）$。对于阶次为2的幂，我们给出了精确数量的嵌套式完美项链，并给出了一个基于矩阵的显式方法来构造每条项链。的计算$ñ$由嵌套式完美项链构建的实数的二进制扩展的第一个数字需要 $Ø（日志ñ）$ 基本数学运算。