In number theory, the geometry of numbers is a topic and method arising from the work of Hermann Minkowski, on the relationship between convex sets and lattices in n-dimensional space. Number theory is the branch of Pure mathematics concerned with the properties of Numbers in general and Integers in particular as well as the wider classes Hermann Minkowski ( June 22 1864 – January 12 1909) was a Russian born German Mathematician, of Jewish In Euclidean space, an object is convex if for every pair of points within the object every point on the Straight line segment that joins them is also within the In Mathematics, especially in Geometry and Group theory, a lattice in R n is a Discrete subgroup of In Mathematics, an n -dimensional space is a Topological space whose Dimension is n (where n is a fixed Natural It has frequently been used in an auxiliary role in proofs, particularly in diophantine approximation. In Number theory, the field of Diophantine approximation, named after Diophantus of Alexandria, deals with the approximation of Real numbers by Rational The subject was given a great deal of attention in the period 1930-1960 by some leading number theorists (including Louis Mordell, Harold Davenport and Carl Ludwig Siegel). Number theory is the branch of Pure mathematics concerned with the properties of Numbers in general and Integers in particular as well as the wider classes Louis Joel Mordell ( 28 January 1888 - 12 March 1972) was a British mathematician known for pioneering research in Number theory. Harold Davenport ( 30 October 1907 – 9 June 1969) was an English mathematician known for his extensive work in Number theory Carl Ludwig Siegel ( December 31 1896 &ndash April 4 1981) was a Mathematician specialising in Number theory.
Minkowski's theorem establishes a relation between symmetric convex sets and integer points; we might as well say, between any lattice and any Banach space norm in n dimensions. In Mathematics, Minkowski's theorem is the statement that any Convex set in R n which is symmetric with respect to the origin and with In Mathematics, the n -dimensional integer lattice (or cubic lattice) denoted Z n, is the lattice in the In Mathematics, Banach spaces (ˈbanax named after Polish Mathematician Stefan Banach) are one of the central objects of study in Functional analysis The topic therefore belongs properly to a sort of affine geometry simplification of the theory of quadratic forms (Hilbert space norms in relation to lattices). Affine geometry is a form of Geometry featuring the unique parallel line property (see the parallel postulate) but where the notion of angle is undefined and lengths In Mathematics, a quadratic form is a Homogeneous polynomial of degree two in a number of variables This article assumes some familiarity with Analytic geometry and the concept of a limit. To relax the convexity technique in a non-trivial way may be technically difficult.
The theoretical foundations can be considered as dealing with the space of lattices in n dimensions, which is a priori the coset space GLn(R)/GLn(Z). This is not easy to deal with directly (it is an example for the theory rather of arithmetic groups). In Mathematics, an arithmetic group ( arithmetic subgroup) in a Linear algebraic group G defined over a Number field K is One foundational result is Mahler's compactness theorem describing the relatively compact subsets (the coset space is non-compact, as can be seen already in the case n = 2, where there are cusps). In Mathematics, Mahler's compactness theorem is a foundational result on lattices in Euclidean space, characterising sets of lattices that are 'bounded'
One can say that the geometry of numbers takes on some of the work that continued fractions do, for diophantine approximation questions in two or more dimensions — there is no straightforward generalisation. In Mathematics, a continued fraction is an expression such as x = a_0 + \cfrac{1}{a_1 + \cfrac{1}{a_2 + \cfrac{1}{a_3 + \cfrac{1}{\ddots\}}}}
References
- Siegel, Carl Ludwig (1989). Carl Ludwig Siegel ( December 31 1896 &ndash April 4 1981) was a Mathematician specialising in Number theory. Lectures on the Geometry of Numbers. Springer-Verlag.
- Hancock, Harris (1939). Development of the Minkowski Geometry of Numbers. Macmillan. (Republished in 1964 by Dover. )
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