A metrical lower bound on the star discrepancy of digital sequences

2013/02/18 by Larcher, Gerhard, Pillichshammer, Friedrich
#11K06 #11K38 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1302.4251

Abstract

In this paper we study uniform distribution properties of digital sequences over a finite field of prime order. In 1998 it was shown by Larcher that for almost all s-dimensional digital sequences the star discrepancy D_N^∗ satisfies an upper bound of the form D_N^∗=O((log N)^s (log log N)^2+ε) for any ε>0. Generally speaking it is much more difficult to obtain good lower bounds for specific sequences than upper bounds. Here we show that Larchers result is best possible up to some log log N term. More detailed, we prove that for almost all s-dimensional digital sequences the star discrepancy satisfies D_N^∗ ≥ c(q,s) (log N)^s log log N for infinitely many N ∈ \NN, where c(q,s)>0 only depends on q and s but not on N.

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