Editing Kronecker's theorem

{{Short description|Theorem about Diophantine approximations}}
{{for| the theorem about the real analytic Eisenstein series|Kronecker limit formula}}
In [[mathematics]], '''Kronecker's theorem''' is a theorem about diophantine approximation,  introduced by {{harvs|txt|authorlink= Leopold Kronecker|first=Leopold|last= Kronecker|year=1884}}.

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 about [[Diophantine approximation]]s that generalizes  [[Dirichlet's approximation theorem]] to multiple variables.

The Kronecker approximation theorem is classically  formulated as follows. 

:''Given real ''n''-[[tuple]]s <math>\alpha_i=(\alpha_{i 1},\dots,\alpha_{i n})\in\mathbb{R}^n, i=1,\dots,m </math> and <math>\beta=(\beta_1,\dots,\beta_n)\in \mathbb{R}^n</math> , the condition: ''
::<math>\forall \epsilon > 0 \, \exists q_i, p_j \in \mathbb Z : \biggl| \sum^m_{i=1}q_i\alpha_{ij}-p_j-\beta_j\biggr|<\epsilon,  1\le j\le n</math> 
:''holds if and only if for any <math>r_1,\dots,r_n\in\mathbb{Z},\ i=1,\dots,m</math> with''
::<math>\sum^n_{j=1}\alpha_{ij}r_j\in\mathbb{Z}, \ \ i=1,\dots,m\ ,</math>
:''the number <math>\sum^n_{j=1}\beta_jr_j</math> is also an integer.''

In plainer language, the first condition states that the tuple <math>\beta = (\beta_1, \ldots, \beta_n)</math> can be approximated arbitrarily well by linear combinations of the <math>\alpha_i</math>s (with integer coefficients) and integer vectors.

For the case of a <math>m=1</math> and <math>n=1</math>, Kronecker's theorem can be stated as follows.<ref>{{Cite web| url=http://mathworld.wolfram.com/KroneckersApproximationTheorem.html| title=Kronecker's Approximation Theorem| publisher=Wolfram Mathworld| language=en| access-date=2019-10-26| archive-date=2018-10-24| archive-url=https://web.archive.org/web/20181024123239/http://mathworld.wolfram.com/KroneckersApproximationTheorem.html| url-status=live}}</ref> For any <math>\alpha, \beta, \epsilon \in \mathbb{R}</math> with <math>\alpha</math> irrational and <math>\epsilon > 0</math> there exist integers <math>p</math> and <math>q</math> with <math>q>0</math>, such that 
::<math>|\alpha q - p - \beta| < \epsilon.</math>

==Relation to tori==

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

:''T'' = ''R<sup>N</sup>/Z<sup>N</sup>'',

the [[closure (mathematics)|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 ''x<sub>i</sub>'' together with 1 should be [[linearly independent]] over the [[rational number]]s, is also [[sufficient condition|sufficient]]. Here it is easy to see that if some [[linear combination]] of the ''x<sub>i</sub>'' and 1 with non-zero rational number coefficients is zero, then the coefficients may be taken as integers, and a [[character (mathematics)|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 (group theory)|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 semigroup|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 ==

*{{citation|last=Kronecker|first= L.
|title=Näherungsweise ganzzahlige Auflösung linearer Gleichungen
|journal=Berl. Ber.|year= 1884|pages= 1179–1193, 1271–1299|url=https://archive.org/stream/n1werkehrsgaufvera03kronuoft#page/46 }}
*{{eom|first=A.L.|last= Onishchik|id=k/k055910|title=Kronecker theorem}}
<references/>

[[Category:Diophantine approximation]]
[[Category:Topological groups]]