# Estimates for logarithmic and Riesz energies for spherical $t$-designs

## TL;DR

This paper derives asymptotic formulas for the discrete logarithmic energy and order estimates for Riesz s-energy of well-separated spherical t-designs on high-dimensional spheres, advancing understanding of their energy properties.

## Contribution

It provides the first asymptotic equalities and order estimates for energies of spherical t-designs, linking design properties with energy asymptotics.

## Key findings

- Asymptotic equalities for logarithmic energy of spherical t-designs
- Order estimates for Riesz s-energy for s ≥ d
- Results applicable to high-dimensional spheres

## Abstract

In this paper we find asymptotic equalities for the discrete logarithmic energy of sequences of well separated spherical $t$-designs on the unit sphere ${\mathbb{S}^{d}\subset\mathbb{R}^{d+1}}$, $d\geq2$. Also we establish exact order estimates for discrete Riesz $s$-energy, $s\geq d$, of sequences of well separated spherical $t$-designs.