Exact order of extreme L p discrepancy of infinite sequences in arbitrary dimension.

We study the extreme L p discrepancy of infinite sequences in the d -dimensional unit cube, which uses arbitrary sub-intervals of the unit cube as test sets. This is in contrast to the classical star L p discrepancy, which uses exclusively intervals that are anchored in the origin as test sets. We show that for any dimension d and any p > 1 , the extreme L p discrepancy of every infinite sequence in [ 0 , 1 ) d is at least of order of magnitude ( log N ) d / 2 , where N is the number of considered initial terms of the sequence. For p ∈ ( 1 , ∞ ) , this order of magnitude is best possible.