For each N>c_d t^d, there exists an N-point spherical t-design in the sphere S^d

You need to know: Notation ${\mathbb N}$ for the set of positive integers, Euclidean space ${\mathbb R}^n$ , sphere in ${\mathbb R}^n$ , d-dimensional Lebesgue measure $\mu_d$ , integration over $\mu_d$ .

Background: Let $S^d$ be a sphere in ${\mathbb R}^{d+1}$ normalised such that $\mu_d(S^d) = 1$ . A set of points $x_1, \dots, x_N \in S^d$ is called a spherical t-design if equality $\int_{S^d} P(x) d \mu_d(x) = \frac{1}{N}\sum\limits_{i=1}^N P(x_i)$ holds for all polynomials P in $d+1$ variables, of total degree at most t.

The Theorem: On 22nd September 2010, Andriy Bondarenko, Danylo Radchenko, and Maryna Viazovska submitted to arxiv a paper in which they proved that for every $d\in {\mathbb N}$ there exist a constant $c_d>0$ such that for each $t\in {\mathbb N}$ and each $N\geq c_d t^d$ , there exists a spherical t-design in $S^d$ consisting of N points.

Short context: The concept of a spherical design was introduced by Delsarte, Goethals, and Seidel in 1977, and, as follows from the definition, it is useful for evaluating integrals. Of course, the smaller N, the easier to compute $\frac{1}{N}\sum\limits_{i=1}^N P(x_i)$ . In 1993, Korevaar and Meyers conjectured the existence of spherical t-designs in $S^d$ with as little as $c_d t^d$ points, which is optimal up to the constant factor. The Theorem confirms this conjecture.