latticepoint

In mathematics, a lattice point is a point in Euclidean space whose coordinates are all integers. In the plane, a lattice point has the form (m, n) with m and n integers. More generally, a lattice point in R^n lies in the integer lattice Z^n, i.e., a point whose coordinates are all integers. The set of lattice points forms a regular grid that partitions space into unit cubes.

The standard lattice Z^n is called the integer lattice. More generally, a lattice in R^n is a discrete subgroup generated by integer combinations of a basis of R^n; equivalently, it can be written as {A k : k in Z^n} for some invertible n by n matrix A. Lattices provide a natural framework for problems involving integer solutions and regular geometric structure, and they underpin many algorithms in number theory and optimization.

Counting lattice points is a central topic in geometry of numbers and discrete geometry. A classic problem asks how many lattice points lie inside or on a given shape, such as a circle or a polygon. The Gauss circle problem studies the number of lattice points in a disk of radius R; asymptotically the count approaches the area, pi R^2, with a fluctuating error term. In higher dimensions, similar questions concern lattice points in polytopes and their dilations; Ehrhart theory describes the counting function for lattice points in scaled polytopes.

Lattice points have applications across mathematics and applied fields. They appear as integer solutions to equations in number theory, as potential atomic positions in crystal structures in crystallography, and as grids used in computer graphics, optimization, and coding theory.

See also: integer lattice, lattice basis, lattice reduction.