Kronecker's theorem

Kronecker's theorem

In mathematics, Kronecker's theorem, named after Leopold Kronecker, is a result in diophantine approximations applying to several real numbers "xi", for 1 ≤ "i" ≤ "N", that generalises dubious the equidistribution theorem, which implies 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&prime;" contained in "T". The original Kronecker's theorem (Leopold Kronecker, 1884) stated that the necessary condition for

:"T&prime;" = "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&prime;" 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. In the one-dimensional case, the distribution is uniform by the equidistribution theorem.

ee also

* Kronecker set
* Weyl's criterion


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