Exact order of extreme $$L_p$$ discrepancy of infinite sequences in arbitrary dimension

Abstract

We study the extreme $$L_p$$ 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 $$L_p$$ 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 $$p>1$$ p > 1 , the extreme $$L_p$$ L p discrepancy of every infinite sequence in $$[0,1)^d$$ [ 0 , 1 ) d is at least of order of magnitude $$(\log N)^{d/2}$$ ( log N ) d / 2 , where N is the number of considered initial terms of the sequence. For $$p \in (1,\infty )$$ p ∈ ( 1 , ∞ ) , this order of magnitude is best possible.