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Classical discrepancies

Discrepancies measure the deviation of a point set from expected ideal distribution. Given an typeset structure-dimensional sequence typeset structure the typeset structure-dimensional counting function is defined by
typeset structure
for any Cartesian box (product of intervals)
typeset structure.

The extremal discrepancy
typeset structure

The star discrepancy
typeset structure

Both are connected by the relations
typeset structure

typeset structure-discrepancy
typeset structure

We have
typeset structure
where the constant
typeset structure depends only on typeset structure.

Theorem. An typeset structure-dimensional sequence typeset structure is uniformly distributed on typeset structure if and only if
typeset structure,
typeset structure,
typeset structure,
typeset structure holds for all continuous functions typeset structure,
typeset structure holds for all typeset structure, typeset structure.
• the one-dimensional sequence
typeset structure, typeset structure, is uniformly distributed for every integer vector typeset structure.

Cite this web-page as:

Štefan Porubský: Classical Discrepancies.

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