Kronecker's Theorem

Kronecker's theorem may also refer to a result in diophantine approximations applying to several real numbers x_i, for 1 ≤ i ≤ N, that generalises Dirichlet's approximation theorem to multiple variables. 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 x_i 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 x_i 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. In the one-dimensional case, the distribution is uniform by the equidistribution theorem.