# Geometry of numbers

*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 $ n $- 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 $ m( F ) $ of some real function

$$ F ( x) = F ( x _ {1} \dots x _ {n} ). $$

Here $ m( F ) $ is the infimum of the values of $ F( x) $ when $ x $ runs through all the integral points (i.e. points with integer coordinates) that satisfy some supplementary condition (e.g. $ x \neq 0 $). In the most important special cases information on $ m ( F ) $ can be obtained from Minkowski's convex-body theorem, which may be formulated as follows. Let $ F( x) < 1 $ be an $ n $- dimensional convex body of volume $ V _ {F} $ and let $ F( - x) = F( x) $ and $ F ( tx) = tF ( x) $ for $ t \geq 0 $; then

$$ m ( F ) \leq 2V _ {F} ^ {- 1/n } . $$

The quantity $ m ( F ) $ is useful in considering conditions of existence of solutions of the Diophantine inequality (cf. Diophantine approximations)

$$ | F ( x) | \leq c. $$

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 $ m( F, \Lambda ) $ of a distance function (cf. Ray function) $ F $ on a lattice of points $ \Lambda $. The concept of a lattice (of points) is a fundamental one in the geometry of numbers. Let $ a _ {1} \dots a _ {n} $ be linearly independent vectors in an $ n $- dimensional Euclidean space. The set of points

$$ \{ g _ {1} a _ {1} + \dots + g _ {n} a _ {n} \} , $$

when $ g _ {1} \dots g _ {n} $ each run through all the integers in an independent manner, is known as the lattice (of points) $ \Lambda $ with basis $ a _ {1} \dots a _ {n} $ and determinant

$$ d ( \Lambda ) = \ | \mathop{\rm det} ( a _ {1} \dots a _ {n} ) |. $$

Let a distance function $ F = F( x) $ and a lattice $ \Lambda $ with determinant $ d ( \Lambda ) $ be given in $ \mathbf R ^ {n} $. The greatest lower bound

$$ m ( F, \Lambda ) = \ \inf _ {\begin{array}{c} a \in \Lambda \\ a \neq 0 \end{array} } F ( a) $$

of the values of $ F $ over the points $ a \neq 0 $ of $ \Lambda $ is called the minimum of $ F $ on $ \Lambda $( or, more accurately, the homogeneous arithmetical minimum). The greatest lower bound $ m( F, \Lambda ) $, 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

$$ F ( x) < 1. $$

In order to estimate $ m( F, \Lambda ) $ from above one must calculate (or estimate) the constant of Hermite $ \gamma ( F ) $ of the distance function $ F $, defined by

$$ \gamma ( F ) = \ \sup _ \Lambda \frac{m ( F, \Lambda ) }{ d ( \Lambda ) ^ {1/n} } , $$

where the supremum is taken over the set $ \mathbf Z _ {n} $ of all $ n $- dimensional lattices $ \Lambda $. There are relations between $ \gamma ( F ) $, the critical determinant (see below) $ \Delta ( \mathfrak C _ {F} ) $ of the set $ \mathfrak C _ {F} = \{ {x } : {F( x) < 1 } \} $ and (if $ F $ is a convex symmetric distance function) the density $ \theta ( \mathfrak C _ {F} ) $ of the densest lattice packing of the body $ \mathfrak C _ {F} $.

Let a set $ \mathfrak M $ and a lattice $ \Lambda $ with determinant $ d ( \Lambda ) $ be given in $ \mathbf R ^ {n} $. The lattice $ \Lambda $ is called admissible for $ \mathfrak M $, or $ \mathfrak M $- admissible, if $ \mathfrak M $ contains no non-zero points from $ \Lambda $. A set $ \mathfrak M $ with at least one admissible lattice is called a set of finite type; otherwise $ \mathfrak M $ is called a set of infinite type. Let $ \mathfrak M $ be a set of finite type; the infimum

$$ \Delta ( \mathfrak M ) = \inf d ( \Lambda ) $$

of the set of determinants $ d ( \Lambda ) $ of all $ \mathfrak M $- admissible lattices $ \Lambda $ is called the critical determinant $ \Delta ( \mathfrak M ) $ of $ \mathfrak M $. Any $ \mathfrak M $- admissible lattice $ \Lambda $ that satisfies the condition

$$ d ( \Lambda ) = \Delta ( \mathfrak M ) $$

is called a critical lattice of $ \mathfrak M $. For a set $ \mathfrak M $ of infinite type one defines $ \Delta ( \mathfrak M ) = + \infty $.

The calculation of the constant of Hermite $ \gamma ( F ) $ of a distance function $ F $ is reduced to the computation of the critical determinant $ \Delta ( \mathfrak C _ {F} ) $ of the star body $ \mathfrak C _ {F} $ defined by $ F( x) < 1 $:

$$ \gamma ( F ) = \{ \Delta ( \mathfrak C _ {F} ) \} ^ {- 1/n } . $$

The connection between the critical determinant and the density of the densest lattice packing is established by the following theorem of Blichfeldt. Let $ \mathfrak R $ be an arbitrary set, let $ D \mathfrak R $ be the corresponding set of differences (i.e. the set of points $ \xi - \eta $, where $ \xi , \eta \in \mathfrak R $) and let $ \Lambda $ be a lattice. For the arrangement $ \{ \mathfrak R , \Lambda \} $, i.e. for the family of sets $ \{ \mathfrak R + a \} $, where $ a \in \Lambda $, to be a packing it is necessary and sufficient that $ \Lambda $ be $ D \mathfrak R $- admissible.

The density $ \theta ( \mathfrak R ) $ of the densest lattice packing of a bounded Lebesgue-measurable set $ \mathfrak R $ of measure $ V ( \mathfrak R ) $ is defined by

$$ \theta ( \mathfrak R ) = \frac{V ( \mathfrak R ) }{\Delta ( D \mathfrak R ) } . $$

For an arbitrary set $ \mathfrak M $ and a Lebesgue-measurable set $ \mathfrak R $ of measure $ V ( \mathfrak R ) $ that satisfies the condition $ D \mathfrak R \subset \mathfrak M $ the following inequality (another formulation of Blichfeldt's theorem) is valid:

$$ \Delta ( \mathfrak M ) \geq V ( \mathfrak R ). $$

If $ \mathfrak K $ is a convex body that is symmetric with respect to a point $ O $, then

$$ \Delta ( \mathfrak K ) = \frac{V ( \mathfrak K ) }{2 ^ {n} \theta ( \mathfrak K ) } , $$

where $ \theta ( \mathfrak K ) $ is the density of the densest lattice packing of $ \mathfrak K $. This means that in the case of a symmetric distance function $ F $ the computation of $ \gamma ( F ) $ is reduced to the computation of the densest lattice packing of the body $ \mathfrak C _ {F} $ defined by $ F( x) < 1 $.

A very important statement in the geometry of numbers is Minkowski's convex-body theorem. Let $ \mathfrak K $ be a convex body that is symmetric with respect to the coordinate origin and of volume $ V ( \mathfrak K ) $. Then

$$ \tag{1 } \Delta ( \mathfrak K ) \geq \ 2 ^ {- n } V ( \mathfrak K ). $$

In other words, a lattice $ \Lambda $ for which

$$ V ( \mathfrak K ) > \ 2 ^ {n} d ( \Lambda ) $$

has a point distinct from zero in $ \mathfrak K $.

Inequality (1) is known as the Minkowski inequality; it gives an estimate from below for the critical determinant $ \Delta ( \mathfrak K ) $ of a convex body $ \mathfrak K $ that is symmetric with respect to 0. In the general case this estimate cannot be improved. Equality is attained if and only if $ \theta ( \mathfrak K ) = 1 $. Convex bodies $ \mathfrak P $ that satisfy the condition $ \theta ( \mathfrak P ) = 1 $ 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 $ F $ and an arbitrary lattice $ \Lambda $ of determinant $ d ( \Lambda ) $ the following inequality is valid:

$$ m ( F, \Lambda ) \leq \ 2 \left \{ \frac{d ( \Lambda ) }{V ( \mathfrak C _ {F} ) } \right \} ^ {1/n} , $$

where

$$ \mathfrak C _ {F} = \{ {x } : {F( x) < 1 } \} . $$

In particular, for the lattice $ \Lambda _ {0} $ of integral points and the distance function

$$ F ( x) = \max _ {1 \leq i \leq n } \ \left \{ \frac{1}{\beta _ {i} } \left | \sum _ {j = 1 } ^ { n } \alpha _ {ij} x _ {j} \right | \right \} $$

Minkowski's theorem on linear homogeneous forms is valid: Let $ \alpha _ {ij} $, $ \beta _ {i} $ be real numbers, $ i, j = 1 \dots n $; $ \beta _ {i} > 0 $, $ | \mathop{\rm det} ( \alpha _ {ij} ) | = \Delta > 0 $. If

$$ \beta _ {1} \dots \beta _ {n} > \Delta , $$

then there exist integers $ x _ {1} \dots x _ {n} $, not all equal to zero, satisfying the system of linear inequalities

$$ \left | \sum _ {j = 1 } ^ { n } \alpha _ {ij} x _ {j} \right | < \beta _ {i} ,\ \ i = 1 \dots n. $$

Geometry of numbers also studies the successive minima of a distance function on a lattice. Let $ F $ be a distance function, let $ \Lambda $ be a lattice and let there be given an index $ i $, $ 1 \leq i \leq n $; then the infimum of the numbers $ \mu $ for which the set $ F( x) < \mu $ contains at least $ i $ linearly independent points of $ \Lambda $ is said to be the $ i $- th successive minimum $ m _ {i} = m _ {i} ( F, \Lambda ) $ of $ F $ on $ \Lambda $. Here

$$ m _ {1} ( F, \Lambda ) = \ m ( F, \Lambda ); \ \ 0 \leq m _ {1} \leq \dots \leq m _ {n} < + \infty . $$

The estimate

$$ \{ m _ {1} ( F, \Lambda ) \} ^ {n} \frac{\Delta ( \mathfrak C _ {F} ) }{d ( \Lambda ) } \leq 1 $$

is valid. It is more difficult to estimate the magnitude

$$ \delta ( F, \Lambda ) = \ \frac{\Delta ( \mathfrak C _ {F} ) \prod _ {i = 1 } ^ { n } m _ {i} ( F, \Lambda ) }{d ( \Lambda ) } $$

from above; to do this, one must be able to compute, or to estimate from above, the quantity

$$ \alpha ( F) = \sup _ \Lambda \delta ( F, \Lambda ), $$

where the supremum is over all $ n $- dimensional lattices $ \Lambda $. The quantity $ \alpha ( F ) $ is called the anomaly of the distance function $ F $, or the anomaly of the set $ \mathfrak C _ {F} $. The inequality $ \alpha ( F ) \geq 1 $ is valid. The following theorem [4] gives an estimate from above for $ \alpha ( F ) $. Let $ F $ be an $ n $- dimensional distance function with anomaly $ \alpha ( F ) $, then

$$ \alpha ( F ) \leq 2 ^ {( n - 1)/2 } . $$

Examples have been constructed to show that this estimate cannot, generally speaking, be improved.

If $ F $ is a convex symmetric distance function, it has been conjectured (the hypothesis on the anomaly of a convex body) that

$$ \alpha ( F ) = 1. $$

Minkowski's second theorem on a convex body, making precise the first theorem, is valid. If $ F $ is a convex symmetric distance function and if $ \Lambda $ is a lattice, then

$$ V ( \mathfrak C _ {F} ) \prod _ {i = 1 } ^ { n } m _ {i} ( F, \Lambda ) \leq \ 2 ^ {n} d ( \Lambda ), $$

where the convex body $ \mathfrak C _ {F} $ is defined by the condition $ F( x) < 1 $. 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 $ \mathfrak C _ {F} $ to arbitrary sets $ \mathfrak M $[9].

The following statement is an estimate from above of the critical determinant of a given set: For any Lebesgue-measurable set $ \mathfrak M $ of measure $ V ( \mathfrak M ) $,

$$ \tag{2 } \Delta ( \mathfrak M ) \leq V ( \mathfrak M ). $$

If $ \mathfrak M $ is a star body that is symmetric with respect to zero, then

$$ \tag{3 } \Delta ( \mathfrak M ) \leq \frac{V ( \mathfrak M ) }{2 \zeta ( n) } , $$

$$ \zeta ( n) = 1 + { \frac{1}{2 ^ {n} } } + \dots . $$

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 $ f $ be a Lebesgue-integrable function on the $ n $- dimensional Euclidean space $ \mathbf R ^ {n} $, and let $ \mu $ be an invariant measure on the space of lattices $ \Lambda $ with determinant 1. Let $ {\mathcal F} $ be the fundamental domain of this space, then

$$ \frac{1}{\mu ( {\mathcal F} ) } \int\limits _ { F } \left \{ \sum _ {\begin{array}{c} \alpha \in \Lambda \\ \alpha \neq 0 \end{array} } f ( a) \right \} d \mu ( \Lambda ) = \ \int\limits _ {\mathbf R ^ {n} } f ( x) dx. $$

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 $ \Delta ( \mathfrak M ) $ of a given set $ \mathfrak M $ from below and from above yield estimates of $ \gamma ( F ) $ 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 $ \Delta ( \mathfrak M ) $ for a given set $ \mathfrak M $( e.g., for a norm body of a given algebraic number field). If $ \mathfrak C $ 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 $ \mathfrak C $( and hence $ \Delta ( \mathfrak C ) $ 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 $ \mathfrak C $ when the dimension $ n \leq 4 $[4].

Generally speaking, finding $ \Delta ( \mathfrak C ) $ is much more difficult for unbounded star bodies $ \mathfrak C $; this is clear by the isolation phenomenon of homogeneous arithmetical minima, which may be described as follows. Let $ F $ be a distance function in $ \mathbf R ^ {n} $, and let the functional

$$ \mu ( \Lambda ) = \mu ( F, \Lambda ) = \ \frac{m ( F) }{d ( \Lambda ) ^ {1/n} } $$

be given on the set $ {\mathcal L} $ of all lattices $ \Lambda $. The set $ M( F ) $ of possible values of $ \mu ( \Lambda ) $ for all $ \Lambda \in {\mathcal L} $ is called the Markov spectrum of $ F $. One says that $ F $ has the isolation phenomenon if the set $ M( F ) $ has isolated points. The set $ M( F ) $ lies in the interval $ ( 0, \gamma ( F )] $. If the star body $ \mathfrak C _ {F} $, $ F( x) < 1 $, is bounded, then

$$ M ( F ) = ( 0, \gamma ( F )]. $$

For this reason the isolation phenomenon is possible for unbounded star bodies only (cf. [4], Chapt. X). The most intensively studied case is $ n = 2 $,

$$ \tag{4 } F _ {0} ( x) = | x _ {1} x _ {2} | ^ {1/2} . $$

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 $ M( F _ {0} ) $ to the right of $ ( 4/9) ^ {1/4} $ is discrete, and has the form

$$ \tag{5 } \left \{ {\left ( { \frac{9}{4} } - { \frac{1}{Q _ {k} } } \right ) ^ {-} 1/4 } : { k = 1, 2 ,\dots } \right \} . $$

Here $ Q _ {k} $ is an increasing sequence of positive integers with the following property: It is possible to find integers $ R _ {k} $, $ S _ {k} $ such that

$$ Q _ {k} ^ {2} + R _ {k} ^ {2} + S _ {k} ^ {2} = \ 3Q _ {k} R _ {k} S _ {k} ; $$

to each point of the spectrum (5) (the "Markov spectrum" in the narrow sense) there corresponds a unique (up to automorphisms [4]) lattice $ \Lambda _ {k} $. The indefinite form $ \phi _ {k} = x _ {1} x _ {2} $, $ ( x _ {1} , x _ {2} ) \in \Lambda _ {k} $, is sometimes called the Markov form, while the sequence $ \phi _ {1} , \phi _ {2} \dots $ is called a Markov chain. It is also known that to the left of some number $ \mu _ {0} = \mu _ {0} ( F _ {0} ) $ the spectrum $ M( F _ {0} ) $ coincides with the segment $ [ 0, \mu _ {0} ] $. 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 $ F $ be a distance function in $ \mathbf R ^ {n} $, let $ \Lambda $ be a lattice of determinant $ d ( \Lambda ) $ in $ \mathbf R ^ {n} $ and let $ x _ {0} $ be a point in $ \mathbf R ^ {n} $. Consider the quantities

$$ l ( x _ {0} ) = \ l ( F, \Lambda ; x _ {0} ) = \ \inf _ {x \equiv x _ {0} ( \Lambda ) } F ( x), $$

$$ l = l ( F, \Lambda ) = \sup _ {x _ {0} \in \mathbf R ^ {n} } l ( F, \Lambda ; x _ {0} ), $$

where the infimum is over all points of the form $ x + a $, $ a \in \Lambda $, while the supremum is over all points $ x _ {0} \in \mathbf R ^ {n} $. The quantity $ l( F, \Lambda ) $ is called the inhomogeneous arithmetical minimum of $ F $ on $ \Lambda $; this "minimum" need not be attained. $ l( F, \Lambda ) $ is the greatest lower bound of the real numbers $ \lambda > 0 $ having the following property: The arrangement $ \{ \lambda \mathfrak C _ {F} , \Lambda \} $ of the set $ \lambda \mathfrak C _ {F} $, where $ \mathfrak C _ {F} $ satisfies the condition $ F( x) < 1 $, over the lattice $ \Lambda $ is a covering, i.e.

$$ \cup _ {a \in \Lambda } ( \lambda \mathfrak C _ {F} + a) = \mathbf R ^ {n} . $$

For the distance function $ F $ one considers the following analogues of the Hermite constant:

$$ \sigma ( F ) = \ \inf _ \Lambda \ \frac{l ( F, \Lambda ) }{d ( \Lambda ) ^ {1/n} } , $$

$$ \Sigma ( F ) = \sup _ \Lambda \frac{l ( F,\ \Lambda ) }{d ( \Lambda ) ^ {1/n} } , $$

where the infimum (supremum) is over all $ n $- dimensional lattices $ \Lambda $. The quantity $ \Sigma ( F ) $ is usually trivial (cf. [4]); if the set $ \mathfrak C _ {F} $, $ F( X) < 1 $, has a finite volume, then

$$ \Sigma ( F ) = + \infty . $$

However, the inhomogeneous problem is connected with $ \Sigma ( F ) $ in one particular instance of the function $ F $ which is of interest.

The hypothesis on the product of inhomogeneous linear forms may be stated as follows. Let

$$ F _ {n} ( x) = \ | x _ {1} \dots x _ {n} | ^ {1/n} , $$

then

$$ \Sigma ( F _ {n} ) = { \frac{1}{2} } . $$

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, $ \sigma ( F ) $ is more informative than $ \Sigma ( F ) $. It is closely related to the value of the density $ \tau ( \mathfrak C _ {F} ) $ of the most economical covering by the body $ \mathfrak C _ {F} $[7], [10]. In fact, if $ F $ is a distance function and if the set $ \mathfrak C _ {F} $ is bounded, then

$$ \tau ( \mathfrak C _ {F} ) = \ \{ \sigma ( F ) \} ^ {n} V ( \mathfrak C _ {F} ). $$

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 $ F $, which are inequalities connecting the inhomogeneous minimum $ l ( F, \Lambda ) $ with the successive homogeneous minima $ m _ {i} ( F, \Lambda ) $( or with the minima of the reciprocal function $ F ^ { * } $ with respect to the reciprocal lattice $ \Lambda ^ {*} $, etc., see [4]). Example. Let $ F $ be a convex symmetric distance function and let $ F( x) > 0 $ for $ x \neq 0 $; then, for any lattice $ \Lambda $,

$$ { \frac{1}{2} } m _ {n} ( F, \Lambda ) \leq \ l ( F, \Lambda ) \leq \ { \frac{1}{2} } \sum _ {k = 1 } ^ { n } m _ {k} ( F, \Lambda ). $$

There exist generalizations of the geometry of numbers to include spaces more general than $ \mathbf R ^ {n} $ and also to discrete sets more general than $ \Lambda $[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 |

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[6] | L. Fejes Toth, "Lagerungen in der Ebene, auf der Kugel und im Raum" , Springer (1972) Zbl 0229.52009 |

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#### Comments

A distance function $ F $ is a non-negative real-valued function on an $ n $- dimensional Euclidean space such that $ F ( tx) = tF ( x) $ for $ t \geq 0 $. If $ F ( x) = F (- x) $ it is called symmetric, and if $ F ( x + y) \leq F ( x) + F ( y) $ it is called convex. If $ F $ is convex, it is required moreover that $ F( x) = 0 $ for $ x = 0 $ 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 $ \mathbf R ^ {n} $ is a family of sets (called tiles) such that their union covers $ \mathbf R ^ {n} $ and their interiors are mutually disjoint. A Dirichlet–Voronoi tiling is a tiling with as tiles sets of the form

$$ \Gamma ( \Lambda , z) = \ \{ {y } : {| y - z | \leq | y - x | \ \textrm{ for } \textrm{ all } x \in \Lambda } \} ,\ \ z \in \Lambda , $$

where $ \Lambda $ is a discrete point set in $ \mathbf R ^ {n} $. 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

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[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 |

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Geometry of numbers.

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