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, J
nos Bolyai Math. Soc., Budapest, 1999, pp. 10-14.
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 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.
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,
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:
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.