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Discrepancies measure the deviation of a point set from expected ideal distribution. Given an -dimensional sequence
the
-dimensional counting function is defined by
for any Cartesian box (product of intervals) .
The extremal discrepancy
The star discrepancy
Both are connected by the relations
-discrepancy
We have
where the constant depends only on
.
Theorem. An -dimensional sequence
is uniformly distributed on
if and only if
• ,
• ,
• ,
• holds for all continuous functions
,
• holds for all
,
.
• the one-dimensional sequence ,
, is uniformly distributed for every integer vector
.
Cite this web-page as:
Štefan Porubský: Classical Discrepancies.