Discrepancy, separation and Riesz energy of finite point sets on the unit sphere

• Authors: Paul Leopardi
• Venue: Advances in Computational Mathematics
• Year: 2013

For $d \geqslant 2,$ we consider asymptotically equidistributed sequences of $\mathbb S^d$ codes, with an upper bound $\operatorname{\boldsymbol{\delta}}$ on spherical cap discrepancy, and a lower bound Δ on separation. For such sequences, if 0驴s驴d, then the difference between the normalized Riesz s energy of each code, and the normalized s-energy double integral on the sphere is bounded above by $\operatorname{O}\big(\operatorname{\boldsymbol{\delta}}^{1-s/d}\,\Delta^{-s}\,N^{-s/d}\big),$ where N is the number of code points. For well separated sequences of spherical codes, this bound becomes $\operatorname{O}\big(\operatorname{\boldsymbol{\delta}}^{1-s/d}\big).$ We apply these bounds to minimum energy sequences, sequences of well separated spherical designs, sequences of extremal fundamental systems, and sequences of equal area points.