# Optimal asymptotic bounds for spherical designs

@article{Bondarenko2010OptimalAB, title={Optimal asymptotic bounds for spherical designs}, author={Andriy V. Bondarenko and Danylo V. Radchenko and Maryna S. Viazovska}, journal={arXiv: Metric Geometry}, year={2010} }

In this paper we prove the conjecture of Korevaar and Meyers: for each $N\ge c_dt^d$ there exists a spherical $t$-design in the sphere $S^d$ consisting of $N$ points, where $c_d$ is a constant depending only on $d$.

## 136 Citations

### Optimal asymptotic bounds for designs on manifolds

- MathematicsAnalysis & PDE
- 2021

We extend to the case of a $d$-dimensional compact connected oriented Riemannian manifold $\mathcal M$ the theorem of A. Bondarenko, D. Radchenko and M. Viazovska on the existence of $L$-designs…

### Estimates for Logarithmic and Riesz Energies of Spherical t-Designs

- MathematicsSpringer Proceedings in Mathematics & Statistics
- 2020

In this paper we find asymptotic equalities for the discrete logarithmic energy of sequences of well separated spherical $t$-designs on the unit sphere ${\mathbb{S}^{d}\subset\mathbb{R}^{d+1}}$,…

### Asymptotically optimal designs on compact algebraic manifolds

- MathematicsMonatshefte für Mathematik
- 2018

We find t-designs on compact algebraic manifolds with a number of points comparable to the dimension of the space of polynomials of degree t on the manifold. This generalizes results on the sphere by…

### Asymptotically optimal designs on compact algebraic manifolds

- Mathematics
- 2016

We find t-designs on compact algebraic manifolds with a number of points comparable to the dimension of the space of polynomials of degree t on the manifold. This generalizes results on the sphere by…

### Upper Energy Bounds for Spherical Designs of Relatively Small Cardinalities

- MathematicsDiscret. Comput. Geom.
- 2021

Upper bounds for the potential energy of spherical designs of cardinality close to the Delsarte–Goethals–Seidel bound are derived by linear programming with use of the Hermite interpolating polynomial of the potential function in suitable nodes.

### Upper Energy Bounds for Spherical Designs of Relatively Small Cardinalities

- MathematicsDiscrete & Computational Geometry
- 2019

We derive upper bounds for the potential energy of spherical designs of cardinality close to the Delsarte–Goethals–Seidel bound. These bounds are obtained by linear programming with use of the…

### Linear Programming Bounds for Covering Radius of Spherical Designs

- Mathematics, Computer ScienceResults in Mathematics
- 2021

This work applies polynomial techniques to obtain lower and upper bounds on the covering radius of spherical designs as function of their dimension, strength, and cardinality and proposes new upper bounds due to Fazekas and Levenshtein.

### Relation between spherical designs through a Hopf map

- Mathematics
- 2015

Cohn--Conway--Elkies--Kumar [Experiment. Math. (2007)] described that one can construct a family of designs on $S^{2n-1}$ from a design on $\mathbb{CP}^{n-1}$. In this paper, we prove their claim for…

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