Kronecker's theorem

<h2 id="statement">Statement</h2>
Kronecker's theorem is a result in <a href="/facts/Diophantine_approximation/WGVMuWF4">diophantine approximations</a> applying to several <a href="/facts/Real_number/R02gw5Pb">real numbers</a> xi, for 1 ≤ i ≤ n, that generalises <a href="/facts/Dirichlet%27s_approximation_theorem/MA2HlQN6">Dirichlet's approximation theorem</a> to multiple variables.
The classical Kronecker approximation theorem is formulated as follows. 

Given real n-<a href="/facts/Tuple/BI1HIxL8">tuples</a> 
 
 
 
 
 α
 
 i
 
 
 =
 (
 
 α
 
 i
 1
 
 
 ,
 …
 ,
 
 α
 
 i
 n
 
 
 )
 ∈
 
 
 R
 
 
 n
 
 
 ,
 i
 =
 1
 ,
 …
 ,
 m
 
 
 {\displaystyle \alpha _{i}=(\alpha _{i1},\dots ,\alpha _{in})\in \mathbb {R} ^{n},i=1,\dots ,m}
 
 and 
 
 
 
 β
 =
 (
 
 β
 
 1
 
 
 ,
 …
 ,
 
 β
 
 n
 
 
 )
 ∈
 
 
 R
 
 
 n
 
 
 
 
 {\displaystyle \beta =(\beta _{1},\dots ,\beta _{n})\in \mathbb {R} ^{n}}
 
 , the condition:

∀
 ϵ
 >
 0
 
 ∃
 
 q
 
 i
 
 
 ,
 
 p
 
 j
 
 
 ∈
 
 Z
 
 :
 
 
 |
 
 
 
 ∑
 
 i
 =
 1
 
 
 m
 
 
 
 q
 
 i
 
 
 
 α
 
 i
 j
 
 
 −
 
 p
 
 j
 
 
 −
 
 β
 
 j
 
 
 
 
 |
 
 
 <
 ϵ
 ,
 1
 ≤
 j
 ≤
 n
 
 
 {\displaystyle \forall \epsilon >0\,\exists q_{i},p_{j}\in \mathbb {Z} :{\biggl |}\sum _{i=1}^{m}q_{i}\alpha _{ij}-p_{j}-\beta _{j}{\biggr |}<\epsilon ,1\leq j\leq n}

holds if and only if for any 
 
 
 
 
 r
 
 1
 
 
 ,
 …
 ,
 
 r
 
 n
 
 
 ∈
 
 Z
 
 ,
  
 i
 =
 1
 ,
 …
 ,
 m
 
 
 {\displaystyle r_{1},\dots ,r_{n}\in \mathbb {Z} ,\ i=1,\dots ,m}
 
 with

∑
          
            j
            =
            1
          
          
            n
          
        
        
          α
          
            i
            j
          
        
        
          r
          
            j
          
        
        ∈
        
          Z
        
        ,
         
         
        i
        =
        1
        ,
        …
        ,
        m
         
        ,
      
    
    {\displaystyle \sum _{j=1}^{n}\alpha _{ij}r_{j}\in \mathbb {Z} ,\ \ i=1,\dots ,m\ ,}

the number 
 
 
 
 
 ∑
 
 j
 =
 1
 
 
 n
 
 
 
 β
 
 j
 
 
 
 r
 
 j
 
 
 
 
 {\displaystyle \sum _{j=1}^{n}\beta _{j}r_{j}}
 
 is also an integer.
In plainer language, the first condition states that the tuple 
 
 
 
 β
 =
 (
 
 β
 
 1
 
 
 ,
 …
 ,
 
 β
 
 n
 
 
 )
 
 
 {\displaystyle \beta =(\beta _{1},\ldots ,\beta _{n})}
 
 can be approximated arbitrarily well by linear combinations of the 
 
 
 
 
 α
 
 i
 
 
 
 
 {\displaystyle \alpha _{i}}
 
s (with integer coefficients) and integer vectors.
For the case of a 
 
 
 
 m
 =
 1
 
 
 {\displaystyle m=1}
 
 and 
 
 
 
 n
 =
 1
 
 
 {\displaystyle n=1}
 
, Kronecker's Approximation Theorem can be stated as follows.<a class="footnote-ref" id="fnref:1" href="#fn:1">1</a> For any 
 
 
 
 α
 ,
 β
 ,
 ϵ
 ∈
 
 R
 
 
 
 {\displaystyle \alpha ,\beta ,\epsilon \in \mathbb {R} }
 
 with 
 
 
 
 α
 
 
 {\displaystyle \alpha }
 
 irrational and 
 
 
 
 ϵ
 >
 0
 
 
 {\displaystyle \epsilon >0}
 
 there exist integers 
 
 
 
 p
 
 
 {\displaystyle p}
 
 and 
 
 
 
 q
 
 
 {\displaystyle q}
 
 with 
 
 
 
 q
 >
 0
 
 
 {\displaystyle q>0}
 
, such that

|
 
 α
 q
 −
 p
 −
 β
 
 |
 
 <
 ϵ
 .
 
 
 {\displaystyle |\alpha q-p-\beta |<\epsilon .}

<h2 id="relation-to-tori">Relation to tori</h2>
In the case of N numbers, taken as a single N-<a href="/facts/Tuple/BI1HIxL8">tuple</a> and point P of the <a href="/facts/Torus/J9SkaV1W">torus</a>

T = RN/ZN,
the <a href="/facts/Closure_(mathematics)/Ct6r2fJz">closure</a> of the subgroup <P> generated by P will be finite, or some torus T′ contained in T. The original Kronecker's theorem (<a href="/facts/Leopold_Kronecker/T0lMQofb">Leopold Kronecker</a>, 1884) stated that the <a href="/facts/Necessary_condition/G45aDCY9">necessary condition</a> for

T′ = T,
which is that the numbers xi together with 1 should be <a href="/facts/Linearly_independent/8JrHfIIa">linearly independent</a> over the <a href="/facts/Rational_number/VJQmyY05">rational numbers</a>, is also <a href="/facts/Sufficient_condition/G45aDCY9">sufficient</a>. Here it is easy to see that if some <a href="/facts/Linear_combination/CVjL6kuN">linear combination</a> of the xi and 1 with non-zero rational number coefficients is zero, then the coefficients may be taken as integers, and a <a href="/facts/Character_(mathematics)/M46SyYUp">character</a> χ of the group T other than the <a href="/facts/Trivial_character/2o9Fvjuj">trivial character</a> takes the value 1 on P. By <a href="/facts/Pontryagin_duality/fbD3d8kz">Pontryagin duality</a> we have T′ contained in the <a href="/facts/Kernel_(group_theory)/GGSpUPMJ">kernel</a> 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 (<a href="/facts/Antitone/MjQK0YL2">antitone</a>) <a href="/facts/Galois_connection/lwuFXX2d">Galois connection</a> between <a href="/facts/Monogenic_semigroup/XmnoSDj6">monogenic</a> 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 <a href="/facts/Equidistribution_theorem/HeAJT6jl">equidistribution theorem</a>.

<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Weyl%27s_criterion/tnz1PnFM">Weyl's criterion</a></li>
<li><a href="/facts/Dirichlet%27s_approximation_theorem/MA2HlQN6">Dirichlet's approximation theorem</a></li></ul>

<ul><li>Kronecker, L. (1884), <a href="https://archive.org/stream/n1werkehrsgaufvera03kronuoft#page/46">"Näherungsweise ganzzahlige Auflösung linearer Gleichungen"</a>, Berl. Ber.: 1179–1193, 1271–1299</li>
<li>Onishchik, A.L. (2001) [1994], <a href="https://www.encyclopediaofmath.org/index.php?title=Kronecker_theorem">"Kronecker theorem"</a>, <a href="/facts/Encyclopedia_of_Mathematics/WC6mGtPm">Encyclopedia of Mathematics</a>, <a href="/facts/European_Mathematical_Society/B3h7b672">EMS Press</a></li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1">"Kronecker's Approximation Theorem". Wolfram Mathworld. Archived from the original on 2018-10-24. Retrieved 2019-10-26. <a href="http://mathworld.wolfram.com/KroneckersApproximationTheorem.html" target="_blank">http://mathworld.wolfram.com/KroneckersApproximationTheorem.html</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
</ol>

Kronecker's theorem open-in-new

Kronecker's theorem