Optimal Discrete Riesz Energy and Discrepancy

The Riesz s-energy of an N-point configuration in the Euclidean space Rp is defined as the sum of reciprocal s-powers of all mutual distances in this system. In the limit s → 0 the Riesz s-potential 1/rs (r the Euclidean distance) governing the point interaction is replaced with the logarithmic potential log(1/r). In particular, we present a conjecture for the leading term of the asymptotic expansion of the optimal L 2-discrepancy with respect to spherical caps on the unit sphere in Rd+1 which follows from Stolarsky’s invariance principle [Proc. Amer. Math. Soc. 41 (1973)] and the fundamental conjecture for the first two terms of the asymptotic expansion of the optimal Riesz s-energy of N points as N → ∞.