Kronecker's Theorem

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.

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 $\alpha_i=(\alpha_,\cdots,\alpha_)\in\mathbb^n, i=1,\cdots,m$ and $\beta=(\beta_1,\cdots,\beta_n)\in \mathbb^n$ , the condition: ''
::$\forall \epsilon > 0 \, \exists q_i, p_j \in \mathbb Z : \biggl, \sum^m_q_i\alpha_-p_j-\beta_j\biggr, <\epsilon, 1\le j\le n$
:''holds if and only if for any $r_1,\dots,r_n\in\mathbb,\ i=1,\dots,m$ with''
::$\sum^n_\alpha_r_j\in\mathbb, \ \ i=1,\dots,m\ ,$
:''the number $\sum^n_\beta_jr_j$ is also an integer.''

In plainer language the first condition states that the tuple $\beta = (\beta_1, \ldots, \beta_n)$ 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 $\alpha, \beta, \epsilon \in \mathbb$ with $\alpha$ irrational and $\epsilon > 0$ there exist integers $p$ and $q$ with $q>0$, such that
::$, \alpha q - p - \beta, < \epsilon.$

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

*
*{{eom, first=A.L., last= Onishchik, id=k/k055910, title=Kronecker's theorem

Diophantine approximation
Topological groups