Editing Low-discrepancy sequence (section)

==The main conjectures==

'''Conjecture 1.''' There is a constant <math>c_s</math> depending only on the dimension <math>s</math>, such that 
:<math>D_{N}^{*}(x_1,\ldots,x_N)\geq c_s\frac{(\ln N)^{s-1}}{N}</math> 
for any finite point set <math>{x_1,\ldots,x_N}</math>. 

'''Conjecture 2.''' There is a constant <math>c'_s</math> depending only on :<math>s</math>, such that:<math>D_{N}^{*}(x_1,\ldots,x_N)\geq c'_s\frac{(\ln N)^{s}}{N}</math> 

for infinite number of <math>N</math> for any infinite sequence <math>x_1,x_2,x_3,\ldots</math>. 

These conjectures are equivalent. They have been proved for <math>s \leq 2</math> by [[Wolfgang M. Schmidt | W. M. Schmidt]]. In higher dimensions, the corresponding problem is still open. The best-known lower bounds are due to [[Michael Lacey (mathematician)|Michael Lacey]] and collaborators.