Published in Research Communications of the conference held in the memory of Paul Erdos Budapest, Hungary, July 4-11, 1999, ed. A. Sali, M. Simonovits and V.T. Ss, Jnos Bolyai Math. Soc., Budapest, 1999, pp. 10-14.

V.V.Arestov and A.G.Babenko, On Kissing Number in Four Dimensions

Let denote the greatest number of nonoverlapping equal balls that touch another ball of the same radius in Due to A.M. Odlyzko and N.J.A. Sloane [11] the problem of determination of the number (kissing number problem in ) was reduced to investigation of minimal distances of arrangements of points on unit sphere in In this presentation we demonstrate possibilities of the classical Delsarte method for estimating the upper bounds of minimal distances of arrangements of and points on .

Introduction. Let denote the unit sphere in real Euclidean -dimensional space where is the inner product of vectors (points) . For a set containing at least two points is called a spherical -code, if for any two points their inner product satisfies to the condition This condition means that an angular distance (angle) between any pair of distinct points in is greater than or equal to :

The size of is simply the number of points in the set ; we denote it by .

The basic problem for spherical codes can be formulated as follows: for given and find a spherical -code of largest possible size . We denote this maximal size by This problem can be consider in the inverse form: for given and find a spherical code with largest possible minimal angle , which we denote by i.e.

The indicated problem and the kissing number problem are connected by equality (cf. [3])

The known example (cf. [3, Ch.1, §2]) of spherical -code in is the set consisting from vertexes of the regular convex polyhedron with Schläfli symbol ; therefore Odlyzko and Sloane [11] have proved that does not exceed ; thus,

In the articles [9], [10], Levenstein obtained universal upper bound for the quantity Levenstein's result can be reformulated for estimating (cf. [2]); in particular,

For concrete cases stronger estimations were presented by Boyvalenkov, Danev and Bumova [2]; for example, (in our notations) they obtained the inequality

The arrangement of points on with the minimal angle was constructed by Hardin, Sloane and Smith [7]. Thus, the following estimations are fulfilled:

Delsarte method. The method which arose in investigations of Delsarte [4] on upper bounds for packings in some metric spaces is often used for estimating from above the quantities This method was developed and successful applied in the works [5], [8], [11], [9], [10], [2] and in of many other articles; the rich bibliography on the subject is contained in monograph [3].

Let be the system of ultraspherical (Gegenbauer) polynomials which are orthogonal on interval with respect to weight function and normed by condition The mentioned approach is founded on a positive-definite property of these polynomials (if we will consider them as kernels on on the Cartesian product ).

For denote by a set consisting of all continuous functions on which are non-positive on and represented on by series with non-negative coefficients and The fact that is non-empty for all was proved in [8]. Let us put

The following statement is contained in [5], [8].

Theorem A. Let Then

Formulation of results. Further we consider four-dimensional case () only. In this case index is equal to and the polynomials are Chebyshev polynomials of the second kind:

Using technique worked out by authors in [1] we find exact values of a parameter for which function takes values and . For formulating the result, we need the following polynomial:

Theorem 1. In the equation has the unique solution: where is thirteenth on the increase real root of polynomial

Consider the polynomial

Theorem 2. In the equation has the unique solution: where is twelfth on the increase real root of polynomial

Corollary. Among arbitrary resp. points located on there exist two points with the angle between them strictly less than resp. . Thus,

Let us remind also (see (1), (2)) that the answer to the question: "whether is the least angular distance between pairs of points of arbitrary configuration of points on strictly less than degrees?" should give a solution of the kissing number problem in

The theorem 1 implies that in order to find the number it is necessary to use other methods. We hope that application of results and ideas of Paul Erdös, his coauthors and pupils, in particular, concerning properties of distribution of distances of given number points on sphere [6], will help to decide problems of such kind.