Some remarks on a discrepancy in compact groups

In this paper we investigate a discrepancy and a L ² discrepancy on compact groups which were introduced by E. Hlawka and W. Fleischer. First we show that this L ² discrepancy is a generalization of the classical diaphony and can be expressed as a finite double sum. We also give estimations of quadrature errors for smooth functions. Then we prove an inequality of Erdos-Turán type for the discrepancy on compact abelian groups and study this inequality in the case of the torus and the dyadic group.