View source for Integral point

{{TEX|done}}{{MSC|52C07|11H06}}

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|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 an example of a [[lattice of points]].

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 [[#References|[a1]]].

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