Minkovskian grid point theorem

The convex ellipse is too large to contain 0 as the only grid point.

The Minkowski lattice point theorem (after Hermann Minkowski ) makes a geometric statement about the position of lattice points in certain sets. If a convex and restricted set symmetrical about the zero point of the grid exceeds a certain size, it must contain further points of the grid in addition to the zero point.

Statement of the sentence

Let be a lattice in , bounded , convex and symmetrical about the origin. If it then holds , then contains a further grid point in addition to the zero point (and because of the symmetry even two). The volume of the grid is defined as the volume of a " basic mesh ". ${\displaystyle \Gamma }$ ${\displaystyle \mathbb {R} ^{d}}$ ${\displaystyle {C}\subseteq \mathbb {R} ^{d}}$ ${\displaystyle \mathrm {vol} (C)>2^{d}\cdot \mathrm {vol} (\Gamma )}$ ${\displaystyle C}$

example

An example of a regular grid in the is . Since a grid mesh is formed by two unit vectors, the volume of this grid is 1. According to the theorem, there is no subset of that that is bounded, convex and symmetrical to the zero point, has an area greater than 4 and has no other grid point besides the zero point contains. ${\displaystyle \mathbb {R} ^{2}}$ ${\displaystyle \mathbb {Z} \times \mathbb {Z} }$ ${\displaystyle \mathbb {R} ^{2}}$

For squares around the zero point is this such a square area can easily be seen as greater than 4 must have an edge length greater than 2 and thus the eight includes grid points , , . However, Minkowski's theorem holds for every bounded, centrally symmetric, convex set, however irregular it may be. ${\displaystyle (0,\pm 1)}$ ${\displaystyle (\pm 1,0)}$ ${\displaystyle (\pm 1,\pm 1)}$

Applications

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There are a number of applications of the Minkowski grid point theorem, starting with the approximation of real numbers by fractions ( Dirichletscher unit theorem ) to "practical" problems such as the question of how far a bullet will fly in a (regularly planted) forest.

literature

• Armin Leutbecher: Number Theory: An Introduction to Algebra . Springer-Verlag, 1996, p. 261. ISBN 3-540-58791-8 .
• Jürgen Neukirch: Algebraic number theory . Springer-Verlag, 2002, p. 28.
• Stefan Müller-Stach, Jens Piontkowski: Elementary and algebraic number theory . Vieweg-Verlag, 2006, p. 65.
• Francois Fricker: Introduction to grid point theory. Birkhäuser 1982, ISBN 376431236X .
• Hans Opolka, Winfried Scharlau From Fermat to Minkowski , Springer-Verlag, Undergraduate Texts in Mathematics, 1985, Chapter 9, pp. 158f