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Classical discrepancies

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 .

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