Kronecker theorem

Given , , and ; then for any there exist integers , , and , , such that

if and only if for any such that

the number

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 generated by the elements , : The closure is precisely the set of all classes such that, for any numbers with

one has also

(Cf. [2].) Under the assumptions of Kronecker's theorem, this closure is simply . This means that the subgroup of all elements of the form

where , is dense in , while the subgroup of vectors

where , is dense in . Kronecker's theorem can be derived from the duality theory for commutative topological groups (cf. Topological group), [3].

In the case , Kronecker's theorem becomes the following proposition: A class , where , generates as a topological group if and only if the numbers are linearly independent over the field of rational numbers. In particular, the torus 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 are linearly independent over , then the set is dense in . Here denotes the fractional part of (cf. Fractional part of a number). In fact, the set 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)