Gitterpunkter

Gitterpunkter, or lattice points, are points in Euclidean space whose coordinates are all integers. In the plane, the standard lattice is the set Z^2, consisting of all points (m, n) with m and n integers. Lattice points form a regular grid and appear in many areas of mathematics and science, including number theory, geometry, crystallography, and computer graphics.

A central topic is counting lattice points inside or on the boundary of geometric regions. If a region R is scaled by a factor t, the number of lattice points in tR is asymptotically proportional to the area of R, with N(tR) ~ Area(R) t^2 for large t. For simple lattice polygons, Pick's theorem provides a precise relation between area and lattice points: Area(R) = I + B/2 − 1, where I is the number of interior lattice points and B the number of lattice points on the boundary.

Beyond two dimensions, lattice points in higher-dimensional space are studied in lattice point theory. Ehrhart theory counts lattice points in dilations of lattice polytopes, producing counting functions that are polynomials or quasi-polynomials in the dilation factor. Computational methods exist to enumerate lattice points, with complexity depending on dimension.

Applications of lattice points span several fields. They are used in integer programming, crystallography, and digital imaging, and have deep connections to number theory, modular forms, and Diophantine equations. The study of lattice points combines geometric intuition with arithmetic structure, providing a bridge between continuous shapes and discrete counting.