In mathematics, Kronecker's theorem is a theorem about diophantine approximation, introduced by .
Kronecker's approximation theorem had been firstly proved by L. Kronecker in the end of the 19th century. It has been now revealed to relate to the idea of n-torus and Mahler measure since the later half of the 20th century. 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.
Kronecker's theorem is a result in diophantine approximations applying to several real numbers xi, for 1 ≤ i ≤ n, that generalises Dirichlet's approximation theorem to multiple variables.
The classical Kronecker approximation theorem is formulated as follows.
Given real n-tuples
\alphai=(\alphai,...,\alphai)\inRn,i=1,...,m
\beta=(\beta1,...,\betan)\inRn
\forall\epsilon>0\existsqi,pj\inZ:l|
m | |
\sum | |
i=1 |
qi\alphaij-pj-\betajr|<\epsilon,1\lej\len
holds if and only if for any
r1,...,rn\inZ, i=1,...,m
n | |
\sum | |
j=1 |
\alphaijrj\inZ, i=1,...,m ,
the number
n | |
\sum | |
j=1 |
\betajrj
In plainer language, the first condition states that the tuple
\beta=(\beta1,\ldots,\betan)
\alphai
For the case of a
m=1
n=1
\alpha,\beta,\epsilon\inR
\alpha
\epsilon>0
p
q
q>0
|\alphaq-p-\beta|<\epsilon.
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. In the one-dimensional case, the distribution is uniform by the equidistribution theorem.