LeVeque type inequalities and discrepancy estimates for minimal energy configurations on spheres

Let SdSd denote the unit sphere in the Euclidean space Rd+1(d≥1). We develop LeVeque type inequalities for the discrepancy between the rotationally invariant probability measure and the normalized counting measures on SdSd. We obtain both upper bound and lower bound estimates. We then use these inequalities to estimate the discrepancy of the normalized counting measures associated with minimal energy configurations on SdSd.