Integral point

A point in an $n$-dimensional space $\mathbf R^n$ with integer coordinates. In number theory one studies the problem of the number of integral points in certain domains; for example, for $n=2$ in a disc and for $n=3$ in a ball (see Circle problem), and also the problem of conditions for uniform distribution of integral points on surfaces; for example, for $n=3$ on a sphere or on an ellipsoid. The strongest results are obtained by the method of trigonometric sums and by methods of algebraic and geometric number theory.

Comments

Integral points are also called lattice points, since the set $\mathbf Z^n\subset\mathbf R^n$ is also referred to as a lattice.

For more geometric problems and results on lattice points see Geometry of numbers. Lattice points are also of importance in crystallography, coding, numerical analysis, analytic number theory, Diophantine approximation, computational geometry, graph theory, integral geometry, and other areas, see [a1].

References

[a1]	P. Erdös, P.M. Gruber, J. Hammer, "Lattice points" , Longman (1989)
[a2]	P.M. Gruber, C.G. Lekkerkerker, "Geometry of numbers" , North-Holland (1987) pp. Sect. (iv) (Updated reprint)
[a3]	J.H. Conway, N.J.A. Sloane, "Sphere packing, lattices and groups" , Springer (1988)