Discrepancy estimates for point sets on the s-dimensional Sierpińnski carpet

Abstract

In a recent paper Cristea and Tichy introduced several types of discrepancies of point sets on the s-dimensional Sierpí nski carpet and proved various relations between these discrepancies. In the present paper we prove a general lower bound for those discrepancies in terms of N, the cardinality of the point set, and we give a probabilistic proof for the existence of point sets with "small" discrepancy. Furthermore we consider a van der Corput type construction of point sets on C _s and determine the exact order of convergence of various notions of discrepancy. Finally, Carpet-Walsh functions are defined to prove an Erdős-Turán-Koksma inequality which we apply to digital point sets on the carpet.