A lower bound for the b-adic diaphony

Abstract

The b-adic diaphony is a quantitative measure for the irregularity of distribution of a point set in the s-dimensional unit cube. In this note we show that the b-adic diaphony (for prime b) of a point set consisting of N points in the s-dimensional unit cube is always at least of order (log N)^{(s−1)/2}/N. This lower bound is best possible.

Keywords

• b-adic diaphony
• l2 discrepancy
• uniform distribution of sequences