ISSN 0032-9460, Problems of Information Transmission, 2012, Vol. 48, No. 3, pp. 283–296.
Pleiades Publishing, Inc., 2012.
Original Russian Text
V.A. Malyshev, 2012, published in Problemy Peredachi Informatsii, 2012, Vol. 48, No. 3, pp. 96–110.
Fine Structure of a One-Dimensional
Discrete Point System
V. A. Malyshev
Laboratory of Large Random Systems, Faculty of Mathematics and Mechanics,
Lomonosov Moscow State University
Received November 30, 2011; in ﬁnal form, May 28, 2012
Abstract—We consider a system of N points x
< ... < x
on a segment of the real line.
An ideal system (crystal) is a system where all distances between neighbors are the same.
Deviation from idealness is characterized by a system of ﬁnite diﬀerences ∇
, for all possible i and k. We ﬁnd asymptotic estimates as N →∞, k →∞,
for a system of points minimizing the potential energy of a Coulomb system in an external ﬁeld.
Assume that a system
0 ≤ x
<... < x
of N diﬀerent points on the segment [0, 1] ∈ R is given. If this system is random, there are a lot
of ways to characterize its structure, for example, as a random-increment process Δ
If there is no any randomness, then there is no conventional way to characterize its organization.
One of possible ways is to consider the system of ﬁnite diﬀerences of several orders =1, 2,...
, ..., ∇
as natural local characteristics of this system of points (or numbers), even if we would not know
the metric of the space where they are embedded or even would not know the space itself. This
corresponds to a situation in calculus where existence of derivatives of suﬃciently large orders
indicates the “quality” of a function. In the discrete case there is no existence problem (for example,
on a circle diﬀerences of arbitrary orders are well deﬁned), but a possible substitute for the existence
can be the decay rate of ﬁnite diﬀerences in (for suﬃciently large N). To go further, everything
depends on the way of how this systems of points is deﬁned. The most popular way is discretization
of smooth functions. For a long time discrete diﬀerences have been used in numerical mathematics
in connection with approximation, interpolation, solution of diﬀerential equations, etc. However,
in these disciplines one does not need diﬀerences of high orders. For arbitrary orders there exists a
classical science, the theory of ﬁnite diﬀerences (see [1–4]), worked out as early as at Newton’s time
(Newton series, divided diﬀerences, etc.). In Section 2 we give necessary deﬁnitions and results
from this science.
We are interested in the cases where there is no natural smooth function whose discretization
is the given point system. Such systems appear, for example, as ﬁxed points of natural dynamical
systems in physics.
We suggest a new insight into a system of diﬀerences as indicators of the deviation scale from
an ideal system. We call a system an ideal system if all distances between neighbors are the same.