Difference between revisions of "Geometry of numbers"
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g044350224.png" /></td> </tr></table> | <table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g044350224.png" /></td> </tr></table> | ||
− | to each point of the spectrum (5) (the | + | to each point of the spectrum (5) (the "Markov spectrum" in the narrow sense) there corresponds a unique (up to automorphisms [[#References|[4]]]) lattice <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g044350225.png" />. The indefinite form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g044350226.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g044350227.png" />, is sometimes called the Markov form, while the sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g044350228.png" /> is called a Markov chain. It is also known that to the left of some number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g044350229.png" /> the spectrum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g044350230.png" /> coincides with the segment <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g044350231.png" />. The isolation phenomenon can be described in terms of admissible lattices (cf. [[#References|[9]]]), which generalizes this concept somewhat. |
The inhomogeneous problem comprises the inhomogeneous Diophantine problems which play an important role in number theory; it forms an important branch of the geometry of numbers. | The inhomogeneous problem comprises the inhomogeneous Diophantine problems which play an important role in number theory; it forms an important branch of the geometry of numbers. | ||
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g044350240.png" /></td> </tr></table> | <table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g044350240.png" /></td> </tr></table> | ||
− | where the infimum is over all points of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g044350241.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g044350242.png" />, while the supremum is over all points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g044350243.png" />. The quantity <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g044350244.png" /> is called the inhomogeneous arithmetical minimum of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g044350245.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g044350246.png" />; this | + | where the infimum is over all points of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g044350241.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g044350242.png" />, while the supremum is over all points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g044350243.png" />. The quantity <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g044350244.png" /> is called the inhomogeneous arithmetical minimum of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g044350245.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g044350246.png" />; this "minimum" need not be attained. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g044350247.png" /> is the greatest lower bound of the real numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g044350248.png" /> having the following property: The arrangement <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g044350249.png" /> of the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g044350250.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g044350251.png" /> satisfies the condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g044350252.png" />, over the lattice <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g044350253.png" /> is a [[Covering|covering]], i.e. |
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g044350254.png" /></td> </tr></table> | <table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044350/g044350254.png" /></td> </tr></table> | ||
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====References==== | ====References==== | ||
− | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> | + | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> H. Minkowski, "Geometrie der Zahlen" , Chelsea, reprint (1953) {{MR|0249269}} {{ZBL|0050.04807}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> H. Minkowski, "Diophantische Approximationen" , Chelsea, reprint (1957) {{MR|0086102}} {{ZBL|53.0165.01}} {{ZBL|38.0220.15}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> H. Hancock, "Development of the Minkowski geometry of numbers" , Macmillan (1939) {{MR|0000400}} {{ZBL|0060.11206}} {{ZBL|65.1156.02}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> J.W.S. Cassels, "An introduction to the geometry of numbers" , Springer (1959) {{MR|0157947}} {{ZBL|0086.26203}} </TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> P.M. Gruber, C.G. Lekkerkerker, "Geometry of numbers" , North-Holland (1987) (Updated reprint) {{MR|0893813}} {{ZBL|0611.10017}} </TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top"> L. Fejes Toth, "Lagerungen in der Ebene, auf der Kugel und im Raum" , Springer (1972) {{MR|}} {{ZBL|0229.52009}} </TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top"> C.A. Rogers, "Packing and covering" , Cambridge Univ. Press (1964) {{MR|0172183}} {{ZBL|0176.51401}} </TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top"> O.-H. Keller, "Geometrie der Zahlen" , ''Enzyklopaedie der math. Wissenschaften mit Einschluss ihrer Anwendungen'' , '''12''' (1954) (Heft 11, Teil III) {{MR|0065595}} {{ZBL|0055.27701}} </TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top"> E. Hlawka, "Grundbegriffe der Geometrie der Zahlen" ''Jahresber. Deutsch. Math.-Verein'' , '''57''' (1954) pp. 37–55 {{MR|0063409}} {{ZBL|0056.27303}} </TD></TR><TR><TD valign="top">[10]</TD> <TD valign="top"> E.P. Baranovskii, "Packings, coverings, partitionings and certain other distributions in spaces of constant curvature" ''Progress Math.'' , '''9''' (1971) pp. 209–253 ''Itogi Nauk. Algebra Topol. Geom. 1967'' (1969) pp. 189–225</TD></TR><TR><TD valign="top">[11]</TD> <TD valign="top"> J.F. Koksma, "Diophantische Approximationen" , Springer (1936) {{MR|0344200}} {{MR|0004857}} {{MR|1545368}} {{ZBL|0012.39602}} {{ZBL|62.0173.01}} </TD></TR><TR><TD valign="top">[12]</TD> <TD valign="top"> A.M. Macbeath, C.A. Rogers, "Siegel's mean value theorem in the geometry of numbers" ''Proc. Cambridge Philos. Soc. (2)'' , '''54''' (1958) pp. 139–151 {{MR|103183}} {{ZBL|}} </TD></TR><TR><TD valign="top">[13]</TD> <TD valign="top"> W. Schmidt, "On the Minkowski–Hlawka theorem" ''Illinois J. Math.'' , '''7''' (1963) pp. 18–23; 714 {{MR|0154828}} {{MR|0146149}} {{ZBL|}} </TD></TR><TR><TD valign="top">[14]</TD> <TD valign="top"> A.M. Markov, "On binary quadratic forms with positive determinant" ''Uspekhi Mat. Nauk'' , '''3''' : 5 (1948) pp. 7–51 (In Russian) {{MR|0027019}} {{ZBL|}} </TD></TR><TR><TD valign="top">[15]</TD> <TD valign="top"> K. Rogers, H.P.F. Swinnerton-Dyer, "The geometry of numbers over algebraic number fields" ''Trans. Amer. Math. Soc.'' , '''88''' (1958) pp. 227–242 {{MR|0095160}} {{ZBL|0083.26206}} </TD></TR></table> |
Line 242: | Line 242: | ||
====References==== | ====References==== | ||
− | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> | + | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> P.M. Gruber, "Geometry of numbers" J. Tölke (ed.) J.M. Wills (ed.) , ''Contributions to geometry'' , Birkhäuser (1979) pp. 186–225 {{MR|0568499}} {{ZBL|0425.10035}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> J.H. Conway, N.J.A. Sloane, "Sphere packing, lattices and groups" , Springer (1987) {{MR|1662447}} {{MR|1194619}} {{MR|1148592}} {{MR|1148591}} {{MR|1541550}} {{MR|0920369}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> P. Erdös, P.M. Gruber, J. Hammer, "Lattice points" , Longman (1989) {{MR|1003606}} {{ZBL|0683.10025}} </TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top"> S.S. Ryskov, "The geometry of positive quadratic forms" , Amer. Math. Soc. (1982) {{MR|0563100}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top"> A.B. Malyshev, Yu.G. Teterina, "Investigations in number theory" , '''9''' , Leningrad (1986) (In Russian)</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top"> T.M. Thompson, "From error-correcting codes through sphere packing to simple groups" , Math. Assoc. Amer. (1983) {{MR|0749038}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top"> G. Fejes Toth, "New results in the theory of packing and covering" P.M. Gruber (ed.) J.M. Wills (ed.) , ''Convexity and its applications'' , Birkhäuser (1983) pp. 318–359 {{MR|}} {{ZBL|0533.52007}} </TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top"> G.F. Voronoi, "Collected works" , '''1–3''' , Kiev (1952) (In Russian) {{MR|0062686}} {{ZBL|0049.02804}} </TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top"> J.H.H. Chalk, "Algebraic lattices" P.M. Gruber (ed.) J.M. Wills (ed.) , ''Convexity and its applications'' , Birkhäuser (1983) pp. 97–110 {{MR|0731107}} {{ZBL|0518.10034}} </TD></TR><TR><TD valign="top">[a10]</TD> <TD valign="top"> H. Minkowski, "Gesammelte Abhandlungen" , '''1–2''' , Teubner (1911) {{MR|}} {{ZBL|42.0023.03}} </TD></TR></table> |
Revision as of 17:33, 31 March 2012
geometric number theory
The branch of number theory that studies number-theoretical problems by the use of geometric methods. Geometry of numbers in its proper sense was formulated by H. Minkowski in 1896 in his fundamental monograph [1]. The starting point of this science, which subsequently became an independent branch of number theory, is the fact (already noted by Minkowski) that certain assertions which seem evident in the context of figures in an -dimensional Euclidean space have far-reaching consequences in number theory.
A fundamental and typical task of the geometry of numbers is the problem to determine the arithmetical minimum of some real function
![]() |
Here is the infimum of the values of
when
runs through all the integral points (i.e. points with integer coordinates) that satisfy some supplementary condition (e.g.
). In the most important special cases information on
can be obtained from Minkowski's convex-body theorem, which may be formulated as follows. Let
be an
-dimensional convex body of volume
and let
and
for
; then
![]() |
The quantity is useful in considering conditions of existence of solutions of the Diophantine inequality (cf. Diophantine approximations)
![]() |
This is a problem to which many problems in number theory can be reduced. The geometry of quadratic forms (cf. Quadratic form) forms a separate chapter in the geometry of numbers.
Two general types of problems are distinguished in the geometry of numbers: the homogeneous and the inhomogeneous problem.
The homogeneous problem, which forms the subject of most studies in the geometry of numbers, deals with the homogeneous minima of a distance function (cf. Ray function)
on a lattice of points
. The concept of a lattice (of points) is a fundamental one in the geometry of numbers. Let
be linearly independent vectors in an
-dimensional Euclidean space. The set of points
![]() |
when each run through all the integers in an independent manner, is known as the lattice (of points)
with basis
and determinant
![]() |
Let a distance function and a lattice
with determinant
be given in
. The greatest lower bound
![]() |
of the values of over the points
of
is called the minimum of
on
(or, more accurately, the homogeneous arithmetical minimum). The greatest lower bound
, which may or may not be attained, is known to be attained by a bounded star body (cf. Star-like domain), which is defined by the inequality
![]() |
In order to estimate from above one must calculate (or estimate) the constant of Hermite
of the distance function
, defined by
![]() |
where the supremum is taken over the set of all
-dimensional lattices
. There are relations between
, the critical determinant (see below)
of the set
and (if
is a convex symmetric distance function) the density
of the densest lattice packing of the body
.
Let a set and a lattice
with determinant
be given in
. The lattice
is called admissible for
, or
-admissible, if
contains no non-zero points from
. A set
with at least one admissible lattice is called a set of finite type; otherwise
is called a set of infinite type. Let
be a set of finite type; the infimum
![]() |
of the set of determinants of all
-admissible lattices
is called the critical determinant
of
. Any
-admissible lattice
that satisfies the condition
![]() |
is called a critical lattice of . For a set
of infinite type one defines
.
The calculation of the constant of Hermite of a distance function
is reduced to the computation of the critical determinant
of the star body
defined by
:
![]() |
The connection between the critical determinant and the density of the densest lattice packing is established by the following theorem of Blichfeldt. Let be an arbitrary set, let
be the corresponding set of differences (i.e. the set of points
, where
) and let
be a lattice. For the arrangement
, i.e. for the family of sets
, where
, to be a packing it is necessary and sufficient that
be
-admissible.
The density of the densest lattice packing of a bounded Lebesgue-measurable set
of measure
is defined by
![]() |
For an arbitrary set and a Lebesgue-measurable set
of measure
that satisfies the condition
the following inequality (another formulation of Blichfeldt's theorem) is valid:
![]() |
If is a convex body that is symmetric with respect to a point
, then
![]() |
where is the density of the densest lattice packing of
. This means that in the case of a symmetric distance function
the computation of
is reduced to the computation of the densest lattice packing of the body
defined by
.
A very important statement in the geometry of numbers is Minkowski's convex-body theorem. Let be a convex body that is symmetric with respect to the coordinate origin and of volume
. Then
![]() | (1) |
In other words, a lattice for which
![]() |
has a point distinct from zero in .
Inequality (1) is known as the Minkowski inequality; it gives an estimate from below for the critical determinant of a convex body
that is symmetric with respect to 0. In the general case this estimate cannot be improved. Equality is attained if and only if
. Convex bodies
that satisfy the condition
are known as parallelohedra. They play an important role in the geometry of numbers and in mathematical crystallography (cf. Crystallography, mathematical).
All applications of Minkowski's convex-body theorem are based on the fact that for a convex symmetric distance function and an arbitrary lattice
of determinant
the following inequality is valid:
![]() |
where
![]() |
In particular, for the lattice of integral points and the distance function
![]() |
Minkowski's theorem on linear homogeneous forms is valid: Let ,
be real numbers,
;
,
. If
![]() |
then there exist integers , not all equal to zero, satisfying the system of linear inequalities
![]() |
Geometry of numbers also studies the successive minima of a distance function on a lattice. Let be a distance function, let
be a lattice and let there be given an index
,
; then the infimum of the numbers
for which the set
contains at least
linearly independent points of
is said to be the
-th successive minimum
of
on
. Here
![]() |
The estimate
![]() |
is valid. It is more difficult to estimate the magnitude
![]() |
from above; to do this, one must be able to compute, or to estimate from above, the quantity
![]() |
where the supremum is over all -dimensional lattices
. The quantity
is called the anomaly of the distance function
, or the anomaly of the set
. The inequality
is valid. The following theorem [4] gives an estimate from above for
. Let
be an
-dimensional distance function with anomaly
, then
![]() |
Examples have been constructed to show that this estimate cannot, generally speaking, be improved.
If is a convex symmetric distance function, it has been conjectured (the hypothesis on the anomaly of a convex body) that
![]() |
Minkowski's second theorem on a convex body, making precise the first theorem, is valid. If is a convex symmetric distance function and if
is a lattice, then
![]() |
where the convex body is defined by the condition
. Minkowski's second theorem is valid [4] independently of the hypothesis on the anomaly of a convex body.
The concept of successive minima and the fundamental results relevant to it (except for the last-named theorem) can be generalized from star bodies to arbitrary sets
[9].
The following statement is an estimate from above of the critical determinant of a given set: For any Lebesgue-measurable set of measure
,
![]() | (2) |
If is a star body that is symmetric with respect to zero, then
![]() | (3) |
![]() |
All proofs of this theorem include some averaging of some function given on the space of lattices. The most natural proof is given by Siegel's mean-value theorem (see, e.g., [12]). Let be a Lebesgue-integrable function on the
-dimensional Euclidean space
, and let
be an invariant measure on the space of lattices
with determinant 1. Let
be the fundamental domain of this space, then
![]() |
As distinct from the estimate from below (1), estimates (2) and (3) are not the best possible (for more precise estimates see [13]).
Estimates of the critical determinant of a given set
from below and from above yield estimates of
from above and from below, i.e. the solution (in a certain sense) of the homogeneous problem in the geometry of numbers. However, it is often important to know the exact value of the critical determinant
for a given set
(e.g., for a norm body of a given algebraic number field). If
is a given bounded star body, then it is possible, in principle, to find an algorithm which permits one to reduce the problem of finding all critical lattices of
(and hence
as well) to a finite number of ordinary problems on the extrema of certain functions of several variables. However, this algorithm is realizable (in the present state of knowledge) only for convex bodies
when the dimension
[4].
Generally speaking, finding is much more difficult for unbounded star bodies
; this is clear by the isolation phenomenon of homogeneous arithmetical minima, which may be described as follows. Let
be a distance function in
, and let the functional
![]() |
be given on the set of all lattices
. The set
of possible values of
for all
is called the Markov spectrum of
. One says that
has the isolation phenomenon if the set
has isolated points. The set
lies in the interval
. If the star body
,
, is bounded, then
![]() |
For this reason the isolation phenomenon is possible for unbounded star bodies only (cf. [4], Chapt. X). The most intensively studied case is ,
![]() | (4) |
A.N. Korkin and E.I. Zolotarev [14] were the first to note the isolation phenomenon in this case (which was also the first case of the isolation phenomenon ever noted). A.A. Markov (see [14]) proved in 1879 that the part of the spectrum to the right of
is discrete, and has the form
![]() | (5) |
Here is an increasing sequence of positive integers with the following property: It is possible to find integers
,
such that
![]() |
to each point of the spectrum (5) (the "Markov spectrum" in the narrow sense) there corresponds a unique (up to automorphisms [4]) lattice . The indefinite form
,
, is sometimes called the Markov form, while the sequence
is called a Markov chain. It is also known that to the left of some number
the spectrum
coincides with the segment
. The isolation phenomenon can be described in terms of admissible lattices (cf. [9]), which generalizes this concept somewhat.
The inhomogeneous problem comprises the inhomogeneous Diophantine problems which play an important role in number theory; it forms an important branch of the geometry of numbers.
Let be a distance function in
, let
be a lattice of determinant
in
and let
be a point in
. Consider the quantities
![]() |
![]() |
where the infimum is over all points of the form ,
, while the supremum is over all points
. The quantity
is called the inhomogeneous arithmetical minimum of
on
; this "minimum" need not be attained.
is the greatest lower bound of the real numbers
having the following property: The arrangement
of the set
, where
satisfies the condition
, over the lattice
is a covering, i.e.
![]() |
For the distance function one considers the following analogues of the Hermite constant:
![]() |
![]() |
where the infimum (supremum) is over all -dimensional lattices
. The quantity
is usually trivial (cf. [4]); if the set
,
, has a finite volume, then
![]() |
However, the inhomogeneous problem is connected with in one particular instance of the function
which is of interest.
The hypothesis on the product of inhomogeneous linear forms may be stated as follows. Let
![]() |
then
![]() |
Studies on this hypothesis and its analogues account for more than one half of all studies on the inhomogeneous problem in the geometry of numbers (cf. Minkowski hypothesis).
In the general case, is more informative than
. It is closely related to the value of the density
of the most economical covering by the body
[7], [10]. In fact, if
is a distance function and if the set
is bounded, then
![]() |
An important chapter of the inhomogeneous problems in the geometry of numbers is constituted by the so-called transference theorems for a given distance function , which are inequalities connecting the inhomogeneous minimum
with the successive homogeneous minima
(or with the minima of the reciprocal function
with respect to the reciprocal lattice
, etc., see [4]). Example. Let
be a convex symmetric distance function and let
for
; then, for any lattice
,
![]() |
There exist generalizations of the geometry of numbers to include spaces more general than and also to discrete sets more general than
[15], [10].
References
[1] | H. Minkowski, "Geometrie der Zahlen" , Chelsea, reprint (1953) MR0249269 Zbl 0050.04807 |
[2] | H. Minkowski, "Diophantische Approximationen" , Chelsea, reprint (1957) MR0086102 Zbl 53.0165.01 Zbl 38.0220.15 |
[3] | H. Hancock, "Development of the Minkowski geometry of numbers" , Macmillan (1939) MR0000400 Zbl 0060.11206 Zbl 65.1156.02 |
[4] | J.W.S. Cassels, "An introduction to the geometry of numbers" , Springer (1959) MR0157947 Zbl 0086.26203 |
[5] | P.M. Gruber, C.G. Lekkerkerker, "Geometry of numbers" , North-Holland (1987) (Updated reprint) MR0893813 Zbl 0611.10017 |
[6] | L. Fejes Toth, "Lagerungen in der Ebene, auf der Kugel und im Raum" , Springer (1972) Zbl 0229.52009 |
[7] | C.A. Rogers, "Packing and covering" , Cambridge Univ. Press (1964) MR0172183 Zbl 0176.51401 |
[8] | O.-H. Keller, "Geometrie der Zahlen" , Enzyklopaedie der math. Wissenschaften mit Einschluss ihrer Anwendungen , 12 (1954) (Heft 11, Teil III) MR0065595 Zbl 0055.27701 |
[9] | E. Hlawka, "Grundbegriffe der Geometrie der Zahlen" Jahresber. Deutsch. Math.-Verein , 57 (1954) pp. 37–55 MR0063409 Zbl 0056.27303 |
[10] | E.P. Baranovskii, "Packings, coverings, partitionings and certain other distributions in spaces of constant curvature" Progress Math. , 9 (1971) pp. 209–253 Itogi Nauk. Algebra Topol. Geom. 1967 (1969) pp. 189–225 |
[11] | J.F. Koksma, "Diophantische Approximationen" , Springer (1936) MR0344200 MR0004857 MR1545368 Zbl 0012.39602 Zbl 62.0173.01 |
[12] | A.M. Macbeath, C.A. Rogers, "Siegel's mean value theorem in the geometry of numbers" Proc. Cambridge Philos. Soc. (2) , 54 (1958) pp. 139–151 MR103183 |
[13] | W. Schmidt, "On the Minkowski–Hlawka theorem" Illinois J. Math. , 7 (1963) pp. 18–23; 714 MR0154828 MR0146149 |
[14] | A.M. Markov, "On binary quadratic forms with positive determinant" Uspekhi Mat. Nauk , 3 : 5 (1948) pp. 7–51 (In Russian) MR0027019 |
[15] | K. Rogers, H.P.F. Swinnerton-Dyer, "The geometry of numbers over algebraic number fields" Trans. Amer. Math. Soc. , 88 (1958) pp. 227–242 MR0095160 Zbl 0083.26206 |
Comments
A distance function is a non-negative real-valued function on an
-dimensional Euclidean space such that
for
. If
it is called symmetric, and if
it is called convex. If
is convex, it is required moreover that
for
only.
The critical determinant of a lattice is also called the lattice constant. An arrangement is also called a set lattice. The inequality (3) is usually called the Minkowski–Hlawka theorem.
In recent years the geometry of numbers has become more geometric in character. The covering and packing problems have been intensively studied, in particular the ball packing problem with its many relations to other areas such as coding, quantization of data, biology, metallurgy. Tilings have also attracted much interest; in particular Dirichlet–Voronoi (and Delone) tilings, which are of interest, for example, in geography, crystallography and computational geometry.
A tiling, or tesselation, of is a family of sets (called tiles) such that their union covers
and their interiors are mutually disjoint. A Dirichlet–Voronoi tiling is a tiling with as tiles sets of the form
![]() |
where is a discrete point set in
. Cf. [5].
Other modern areas of the geometry of numbers are the theory of the zeta-function on lattices and (computational) reduction theory of quadratic forms and lattices.
References
[a1] | P.M. Gruber, "Geometry of numbers" J. Tölke (ed.) J.M. Wills (ed.) , Contributions to geometry , Birkhäuser (1979) pp. 186–225 MR0568499 Zbl 0425.10035 |
[a2] | J.H. Conway, N.J.A. Sloane, "Sphere packing, lattices and groups" , Springer (1987) MR1662447 MR1194619 MR1148592 MR1148591 MR1541550 MR0920369 |
[a3] | P. Erdös, P.M. Gruber, J. Hammer, "Lattice points" , Longman (1989) MR1003606 Zbl 0683.10025 |
[a4] | S.S. Ryskov, "The geometry of positive quadratic forms" , Amer. Math. Soc. (1982) MR0563100 |
[a5] | A.B. Malyshev, Yu.G. Teterina, "Investigations in number theory" , 9 , Leningrad (1986) (In Russian) |
[a6] | T.M. Thompson, "From error-correcting codes through sphere packing to simple groups" , Math. Assoc. Amer. (1983) MR0749038 |
[a7] | G. Fejes Toth, "New results in the theory of packing and covering" P.M. Gruber (ed.) J.M. Wills (ed.) , Convexity and its applications , Birkhäuser (1983) pp. 318–359 Zbl 0533.52007 |
[a8] | G.F. Voronoi, "Collected works" , 1–3 , Kiev (1952) (In Russian) MR0062686 Zbl 0049.02804 |
[a9] | J.H.H. Chalk, "Algebraic lattices" P.M. Gruber (ed.) J.M. Wills (ed.) , Convexity and its applications , Birkhäuser (1983) pp. 97–110 MR0731107 Zbl 0518.10034 |
[a10] | H. Minkowski, "Gesammelte Abhandlungen" , 1–2 , Teubner (1911) Zbl 42.0023.03 |
Geometry of numbers. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Geometry_of_numbers&oldid=24077