Editing Low-discrepancy sequence (section)

==Lower bounds==

Let <math>s=1</math>. Then

:<math>
D_N^*(x_1,\ldots,x_N)\geq\frac{1}{2N}
</math>

for any finite point set <math>\{x_1, \dots, x_N }\</math>.

Let <math>s=2</math>. [[Wolfgang M. Schmidt|W. M. Schmidt]] proved that for any finite point set <math>\{x_1, \dots, x_N }\</math>,

:<math>
D_N^*(x_1,\ldots,x_N)\geq C\frac{\log N}{N}
</math>

where

:<math>
C=\max_{a\geq3}\frac{1}{16}\frac{a-2}{a\log a}=0.023335\dots.
</math>

For arbitrary dimensions<math>s > 1</math>, [[Klaus Roth|K. F. Roth]] proved that

:<math>
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}
</math>

for any finite point set <math>\{x_1, \dots, x_N }\</math>. 
Jozef Beck <ref>{{cite journal|title=A two-dimensional van Aardenne-Ehrenfest theorem in irregularities of distribution|journal= Compositio Mathematica|volume= 72 |issue=3|year=1989|pages= 269–339|s2cid=125940424|last=Beck|first= József|url=https://eudml.org/doc/89992|mr= 1032337 | zbl= 0691.10041}}</ref> established a double log improvement of this result in three dimensions. 
This was improved by D. Bilyk and [[Michael Lacey (mathematician)| M. T. Lacey]] to a power of a single logarithm. The best known bound for ''s''&nbsp;>&nbsp;2 is due
D. Bilyk and [[Michael Lacey (mathematician)| M. T. Lacey]] and A. Vagharshakyan.<ref>{{Cite journal|doi=10.1016/j.jfa.2007.09.010|title=On the Small Ball Inequality in all dimensions |year=2008 |last1=Bilyk |first1=Dmitriy |last2=Lacey |first2=Michael T. |last3=Vagharshakyan |first3=Armen |journal=Journal of Functional Analysis |volume=254 |issue=9 |pages=2470–2502 |s2cid=14234006 |doi-access=free |arxiv=0705.4619 }}</ref> There exists a <math>t>0</math> depending on ''s'' so that
:<math>
D_N^*(x_1,\ldots,x_N)\geq  t  \frac{\log^{\frac{s-1}{2}+t}N}{N}
</math>

for any finite point set  <math>\{x_1, \dots, x_N }\</math>.