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)

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.