Mh. Math. 128, 179±188 (1999) Discrepancy Estimates on the Sphere By V. V. Andrievskii, H.-P. Blatt, and M. Go Ètz, Katholische Universita Èt Eichsta Ètt, Germany (Received 1 August 1998; in revised form 30 December 1998) Abstract. For measures on the unit sphere in R d , d53, we derive discrepancy estimates in terms of the quality of corresponding quadrature formulas and in terms of bounds for potential differences. 1. Introduction and Statement of the Results Distributing points on the unit sphere in R 3 has attracted the interest of many mathematicians (see [6] and [15] for an interesting overview). Although applications go far beyond the construction of quadrature rules on the sphere, this paper is devoted in part to a relationship between the equidistribution of points and the quality of a corresponding quadrature formula. The other part, namely, discrepancy estimates in terms of bounds for Newtonian potentials, has its roots in the complex plane. Starting from quantitative equidistribution results by ERDO È S and TURA Â N [10] on the unit disk and unit interval for the zeros of polynomials, the second author and GROTHMANN [7] were able to give a potential-theoretic interpretation, which in the sequel led to various discrepancy estimates for the distribution of zeros of polynomials (see, e.g., [21], [8], [1]). We place ourselves in R d , d53, and denote by  the surface measure on the unit sphere S fx 2 R d : jxj 1g, normalized to total mass 1. Here, jj denotes the euclidean norm. We are interested in the question, how well one can approximate  by means of certain masses on S. More speci®cally, suppose  is a unit measure on S. For reasonable classes B of test sets B  S we focus on the discrepancy sup B2B jB Bj: Our choice of B is based on the following de®nition, introduced by SJO È GREN [20]. A measurable set B  S is said to be K -regular, if for @ S  B : y 2 S j dist y; B4; dist y; S n B4 f g it holds that @ S  B4K > 0: 1991 Mathematics Subject Classi®cation: 11K38, 41A55, 31B15 Key words: Discrepancy, Newtonian potential, quadrature formula