Low-discrepancy Sequence - Lower Bounds

Lower Bounds

Let s = 1. Then

D_N^*(x_1,\ldots,x_N)\geq\frac{1}{2N}

for any finite point set {x1, ..., xN}.

Let s = 2. W. M. Schmidt proved that for any finite point set {x1, ..., xN},

D_N^*(x_1,\ldots,x_N)\geq C\frac{\log N}{N}

where

C=\max_{a\geq3}\frac{1}{16}\frac{a-2}{a\log a}=0.023335\dots

For arbitrary dimensions s > 1, K.F. Roth proved that

D_N^*(x_1,\ldots,x_N)\geq\frac{1}{2^{4s}}\frac{1}{((s-1)\log2)^\frac{s-1}{2}}\frac{\log^{\frac{s-1}{2}}N}{N}

for any finite point set {x1, ..., xN}. This bound is the best known for s > 3.

Read more about this topic:  Low-discrepancy Sequence

Famous quotes containing the word bounds:

    “Firmness yclept in heroes, kings and seamen,
    That is, when they succeed; but greatly blamed
    As obstinacy, both in men and women,
    Whene’er their triumph pales, or star is tamed —
    And ‘twill perplex the casuist in morality
    To fix the due bounds of this dangerous quality.”
    —George Gordon Noel Byron (1788–1824)

    “Great Wits are sure to Madness near alli’d
    And thin Partitions do their Bounds divide;
    Else, why should he, with Wealth and Honour blest,
    Refuse his Age the needful hours of Rest?”
    —John Dryden (1631–1700)