Constructions of Low-discrepancy Sequences - The Van Der Corput Sequence

The Van Der Corput Sequence

Let

$n=\sum_{k=0}^{L-1}d_k(n)b^k$

be the b-ary representation of the positive integer n ≥ 1, i.e. 0 ≤ d_k(n) < b. Set

$g_b(n)=\sum_{k=0}^{L-1}d_k(n)b^{-k-1}.$

Then there is a constant C depending only on b such that (g_b(n))_{n ≥ 1} satisfies

$D^*_N(g_b(1),\dots,g_b(N))\leq C\frac{\log N}{N}.$

where D*_N is the star discrepancy.

Read more about this topic: Constructions Of Low-discrepancy Sequences

Famous quotes containing the words van, der and/or sequence:

“The Mediterranean has the color of mackerel, changeable I mean. You don’t always know if it is green or violet, you can’t even say it’s blue, because the next moment the changing reflection has taken on a tint of rose or gray.”
—Vincent Van Gogh (1853–1890)

“Under the lindens on the heather,
There was our double resting-place.”
—Walther Von Der Vogelweide (1170?–1230?)

“Reminiscences, even extensive ones, do not always amount to an autobiography.... For autobiography has to do with time, with sequence and what makes up the continuous flow of life. Here, I am talking of a space, of moments and discontinuities. For even if months and years appear here, it is in the form they have in the moment of recollection. This strange form—it may be called fleeting or eternal—is in neither case the stuff that life is made of.”
—Walter Benjamin (1892–1940)