Kronecker's theorem

Template:For In mathematics, Kronecker's theorem is a theorem about diophantine approximation, introduced by Template:Harvs.

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.

Statement

Kronecker's theorem is 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.

The classical Kronecker approximation theorem is formulated as follows.

Given real n-tuples ${\alpha }_i=({\alpha }_{i1},\dots ,{\alpha }_{in})\in {ℝ}^n,i=1,\dots ,m$ and $\beta =({\beta }_1,\dots ,{\beta }_n)\in {ℝ}^n$ , the condition:

$\mathrm{\forall }ϵ>0\mathrm{\exists }{q}_i,p_j\in \mathbb{Z}:|\sum _{i=1}^m{q}_i{\alpha }_{ij}-p_j-{\beta }_j|<ϵ,1\le j\le n$

holds if and only if for any $r_1,\dots ,r_n\in \mathbb{Z},i=1,\dots ,m$ with

$\sum _{j=1}^n{\alpha }_{ij}{r}_j\in \mathbb{Z},i=1,\dots ,m,$

the number $\sum _{j=1}^n{\beta }_j{r}_j$ is also an integer.

In plainer language, the first condition states that the tuple $\beta =({\beta }_1,\dots ,{\beta }_n)$ can be approximated arbitrarily well by linear combinations of the ${\alpha }_i$s (with integer coefficients) and integer vectors.

For the case of a $m=1$ and $n=1$, Kronecker's Approximation Theorem can be stated as follows. For any $\alpha ,\beta ,ϵ\in ℝ$ with $\alpha$ irrational and $ϵ>0$ there exist integers $p$ and $q$ with $q>0$, such that

$|\alpha q-p-\beta |<ϵ.$

Relation to tori

In the case of N numbers, taken as a single N-tuple and point P of the torus

T = R^N/Z^N,

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.

See also

• Weyl's criterion
• Dirichlet's approximation theorem

References

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