Difference between revisions of "Integral point"
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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. | 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. | ||
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====Comments==== | ====Comments==== | ||
| − | Integral points are also called lattice points, since the set $\mathbf Z^n\subset\mathbf R^n$ is | + | 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 [[ | + | 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==== | ====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> | + | <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> | ||
Latest revision as of 19:03, 2 October 2016
2020 Mathematics Subject Classification: Primary: 52C07 Secondary: 11H06 [MSN][ZBL]
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 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 [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) |
Integral point. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Integral_point&oldid=32542