以10为基数的范德科皮特序列前n项(n从0至999)的图示
范德科皮特序列(英语: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。
使用基数b可将自然数n表示为
![{\displaystyle n=\sum _{k=0}^{L-1}d_{k}(n)b^{k},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/24d41846a7f859c8e454b9d1f3d2b80e2a8f531f)
其中第k位为dk(n),满足0 ≤ dk(n) < b。
由此,可以得到范德科皮特序列的第n位:
![{\displaystyle g_{b}(n)=\sum _{k=0}^{L-1}d_{k}(n)b^{-k-1}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/24663095108966037060fc804baf2f8658bb24f0)
例如,以10为基数的范德科皮特序列的前几项为
![{\displaystyle \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\},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2cbc3b8897cf240466ba218e77c32d9e3649067d)
而以2为基数的范德科皮特序列的前几项则为
![{\displaystyle \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\}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ca8fdc78e5aec5a35f778b3437b038d7084ecf1b)
参考文献[编辑]
- 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