Kronecker's theorem
From Academic Kids
In mathematics, Kronecker's theorem is a result in diophantine approximation applying to several real numbers xi, for 1 ≤ i ≤ N, which generalises the fact that an infinite cyclic subgroup of the unit circle group is a dense subset. In terms of physical systems, it has the consequence that planets in circular orbits moving uniformly around a star will, over time, assume all alignments, unless there is an exact dependency between their orbital periods.
In the case of N numbers, taken as a single N-tuple and point P of the torus
- T = RN/ZN,
the closure of the subgroup <P> generated by P will be finite, or some torus T′ contained in T. The original Kronecker's theorem (Leopold Kronecker, 1884) stated that the necessary condition for
- T′ = T,
which is that the numbers xi together with 1 should be linearly independent over the rational numbers, is also sufficient. Here it is easy to see that if some linear combination of the xi and 1 with non-zero rational number coefficients is zero, then the coefficients may be taken as integers, and a character χ of the group T other than the trivial character takes the value 1 on P. By Pontryagin duality we have T′ contained in the kernel of χ, and therefore not equal to T.
In fact a thorough use of Pontryagin duality here shows that the whole Kronecker theorem describes the closure of <P> as the intersection of the kernels of the χ with
- χ(P) = 1.
This gives an (antitone) Galois connection between monogenic closed subgroups of T (those with a single generator, in the topological sense), and sets of characters with kernel containing a given point. Not all closed subgroups occur as monogenic; for example a subgroup that has a torus of dimension ≥ 1 as connected component of the identity element, and that is not connected, cannot be such a subgroup.
The theorem leaves open the question of how well (uniformly) the multiples mP of P fill up the closure.
See also: Kronecker set
