Kronecker theorem

Given $ a _ {i} = ( a _ {i1} \dots a _ {in} ) \in \mathbf R ^ {n} $, $ i = 1 \dots m $, and $ b = ( b _ {1} \dots b _ {n} ) \in \mathbf R ^ {n} $; then for any $ \epsilon > 0 $ there exist integers $ q _ {i} $, $ i = 1 \dots m $, and $ p _ {j} $, $ j = 1 \dots n $, such that

$$ \left | \sum _ {i = 1 } ^ { m } q _ {i} a _ {ij} - p _ {j} - b _ {j} \ \right | < \epsilon ,\ \ 1 \leq j \leq n, $$

if and only if for any $ r _ {1} \dots r _ {n} \in \mathbf Z $ such that

$$ \sum _ {j = 1 } ^ { n } a _ {ij} r _ {j} \in \mathbf Z ,\ \ i = 1 \dots m, $$

the number

$$ \sum _ {j = 1 } ^ { n } b _ {j } r _ {j } $$

is also an integer. This theorem was first proved in 1884 by L. Kronecker (see [1]).

Kronecker's theorem is a special case of the following theorem [2], which describes the closure of the subgroup of the torus $ T ^ {n} = \mathbf R ^ {n} / \mathbf Z ^ {n} $ generated by the elements $ a _ {i} + \mathbf Z ^ {n} $, $ i = 1 \dots m $: The closure is precisely the set of all classes $ b + \mathbf Z ^ {n} $ such that, for any numbers $ r _ {1} \dots r _ {n} \in \mathbf Z $ with

$$ \sum _ {j = 1 } ^ { n } a _ {ij} r _ {j} \in \mathbf Z ,\ \ i = 1 \dots n, $$

one has also

$$ \sum _ {j = 1 } ^ { n } b _ {j} r _ {j} \in \mathbf Z . $$

(Cf. [2].) Under the assumptions of Kronecker's theorem, this closure is simply $ T ^ {n} $. This means that the subgroup of all elements of the form

$$ \sum _ {i = 1 } ^ { m } q _ {i} ( a _ {i} + \mathbf Z ) ^ {n} , $$

where $ q _ {i} \in \mathbf Z $, is dense in $ T ^ {n} $, while the subgroup of vectors

$$ \sum _ {i = 1 } ^ { m } q _ {i} a _ {i} + p, $$

where $ p \in \mathbf Z ^ {n} $, is dense in $ \mathbf R ^ {n} $. Kronecker's theorem can be derived from the duality theory for commutative topological groups (cf. Topological group), [3].

In the case $ m = 1 $, Kronecker's theorem becomes the following proposition: A class $ \omega + \mathbf Z ^ {n} $, where $ \omega = ( \omega _ {1} \dots \omega _ {n} ) \in \mathbf R ^ {n} $, generates $ T ^ {n} $ as a topological group if and only if the numbers $ 1, \omega _ {1} \dots \omega _ {n} $ are linearly independent over the field $ \mathbf Q $ of rational numbers. In particular, the torus $ T ^ {n} $ as a topological group is monothetic, i.e. is generated by a single element.

References

[1]	L. Kronecker, "Näherungsweise ganzzahlige Auflösung linearer Gleichungen" , Werke , 3 , Chelsea, reprint (1968) pp. 47–109
[2]	N. Bourbaki, "Elements of mathematics. General topology" , Addison-Wesley (1966) (Translated from French)
[3]	L.S. Pontryagin, "Topological groups" , Princeton Univ. Press (1958) (Translated from Russian)

Comments

The last statement above can be rephrased as: If $ \omega _ {1} \dots \omega _ {n} $ are linearly independent over $ \mathbf Q $, then the set $ B= \{ ( \{ k \omega _ {1} \} \dots \{ k \omega _ {n} \} ) : k \in \mathbf Z \} $ is dense in $ ( 0, 1) $. Here $ \{ x \} = x - [ x] $ denotes the fractional part of $ x $( cf. Fractional part of a number). In fact, the set $ B $ is even uniformly distributed, cf. Uniform distribution.

References

[a1]	G.H. Hardy, E.M. Wright, "An introduction to the theory of numbers" , Oxford Univ. Press (1979) pp. Chapt. 23
[a2]	J.W.S. Cassels, "An introduction to diophantine approximation" , Cambridge Univ. Press (1957)