An extension of Delsarte's method. The kissing problem in three and four dimensions

Positive definite functions in distance geometry

I. J. Schoenberg proved that a function is positive definite in the unit sphere if and only if this function is a nonnegative linear combination of Gegenbauer polynomials. This fact play a crucial role in Delsarte's method for finding bounds for the density of sphere packings on spheres and Euclidean spaces. One of the most excited applications of Delsarte's method is a solution of the kissing number problem in dimensions 8 and 24. However, 8 and 24 are the only dimensions in which this method gives a precise result. For other dimensions (for instance, three and four) the upper bounds exceed t…

The kissing problem in three dimensions

The kissing number k(3) is the maximal number of equal size nonoverlapping spheres in three dimensions that can touch another sphere of the same size. This number was the subject of a famous discussion between Isaac Newton and David Gregory in 1694. The first proof that k(3)=12 was given by Schütte and van der Waerden only in 1953. In this paper we present a new solution of the Newton-Gregory problem that uses our extension of the Delsarte method. This proof relies on basic calculus and simple spherical geometry.

The kissing number in four dimensions

The kissing number problem asks for the maximal number k(n) of equal size nonoverlapping spheres in n-dimensional space that can touch another sphere of the same size. This problem in dimension three was the subject of a famous discussion between Isaac Newton and David Gregory in 1694. In three dimensions the problem was finally solved only in 1953 by Schütte and van der Waerden. In this paper we present a solution of a long-standing problem about the kissing number in four dimensions. Namely, the equality k(4)=24 is proved. The proof is based on a modification of Delsarte's method.

Codes in spherical caps

We consider bounds on codes in spherical caps and related problems in geometry and coding theory. An extension of the Delsarte method is presented that relates upper bounds on the size of spherical codes to upper bounds on codes in caps. Several new upper bounds on codes in caps are derived. Applications of these bounds to estimates of the kissing numbers and one-sided kissing numbers are considered. It is proved that the maximum size of codes in spherical caps for large dimensions is determined by the maximum size of spherical codes, so these problems are asymptotically equivalent.

关于同质空间的完整离散Schoenberg-Deslarte理论

Complete discrete Schoenberg-Delsarte theory for homogeneous spaces

本文提出了一种关于同质空间的完整离散Schoenberg-Delsarte理论，扩展了Schoenberg的经典特征化，应用于受限排列问题和Delsarte理论，并证明了特定正定函数必须来自多项式。

具有亲吻常数 $\tau(\mathcal{L}_n) \ge e^{\sqrt{n}}$ 的格族

A lattice family with kissing numbers $τ(\mathcal{L}_n) \ge e^{2 \sqrt{n}}$

本文提出一种素数维度整数格族，其亲吻常数下界达到 $e^{\sqrt{n}}$ 量级，优于 Barnes-Wall 格，基于 Reed-Solomon 码推广构造。