范德科皮特序列

范德科皮特序列（英語：van der Corput sequence）是定义在单位区间上的一维低差异序列，由荷兰数学家约翰内斯·范德科皮特于1935年提出。将以基数b表示的自然数列反转后便可得到范德科皮特序列。

以10为基数的范德科皮特序列前n项（n从0至999）的图示

使用基数b可将自然数n表示为

$n=\sum _{k=0}^{L-1}d_{k}(n)b^{k},$

其中第k位为d_k(n)，满足0 ≤ d_k(n) < b。

由此，可以得到范德科皮特序列的第n位：

$g_{b}(n)=\sum _{k=0}^{L-1}d_{k}(n)b^{-k-1}.$

例如，以10为基数的范德科皮特序列的前几项为

$\left\{{\tfrac {1}{10}},{\tfrac {2}{10}},{\tfrac {3}{10}},{\tfrac {4}{10}},{\tfrac {5}{10}},{\tfrac {6}{10}},{\tfrac {7}{10}},{\tfrac {8}{10}},{\tfrac {9}{10}},{\tfrac {1}{100}},{\tfrac {11}{100}},{\tfrac {21}{100}},{\tfrac {31}{100}},{\tfrac {41}{100}},{\tfrac {51}{100}},{\tfrac {61}{100}},{\tfrac {71}{100}},{\tfrac {81}{100}},{\tfrac {91}{100}},{\tfrac {2}{100}},{\tfrac {12}{100}},{\tfrac {22}{100}},{\tfrac {32}{100}},\ldots \right\},$

而以2为基数的范德科皮特序列的前几项则为

$\left\{{\tfrac {1}{2}},{\tfrac {1}{4}},{\tfrac {3}{4}},{\tfrac {1}{8}},{\tfrac {5}{8}},{\tfrac {3}{8}},{\tfrac {7}{8}},{\tfrac {1}{16}},{\tfrac {9}{16}},{\tfrac {5}{16}},{\tfrac {13}{16}},{\tfrac {3}{16}},{\tfrac {11}{16}},{\tfrac {7}{16}},{\tfrac {15}{16}},\ldots \right\}.$

参考文献

• van der Corput, J.G., Verteilungsfunktionen. I. Mitt., Proc. Akad. Wet. Amsterdam, 1935, 38: 813–821, Zbl 0012.34705 （德语） 
• Kuipers, L.; Niederreiter, H., Uniform distribution of sequences, Dover Publications: 129,158, 2005 [1974], ISBN 0-486-45019-8, Zbl 0281.10001