Kronecker's theorem

{{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 approximations that generalizes Dirichlet's approximation theorem to multiple variables.

The Kronecker approximation theorem is classically formulated as follows.

:Given real n-tuples $\alpha_i=(\alpha_{i 1},\dots,\alpha_{i n})\in\mathbb{R}^n, i=1,\dots,m$ and $\beta=(\beta_1,\dots,\beta_n)\in \mathbb{R}^n$ , the condition:

:: $\forall \epsilon > 0 \, \exists q_i, p_j \in \mathbb Z : \biggl| \sum^m_{i=1}q_i\alpha_{ij}-p_j-\beta_j\biggr|<\epsilon, 1\le j\le n$

:holds if and only if for any $r_1,\dots,r_n\in\mathbb{Z},\ i=1,\dots,m$ with

:: $\sum^n_{j=1}\alpha_{ij}r_j\in\mathbb{Z}, \ \ i=1,\dots,m\ ,$

:the number $\sum^n_{j=1}\beta_jr_j$ is also an integer.

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

For the case of a $m=1$ and $n=1$ , Kronecker's theorem can be stated as follows.{{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}} For any $\alpha, \beta, \epsilon \in \mathbb{R}$ with $\alpha$ irrational and $\epsilon > 0$ there exist integers $p$ and $q$ with $q>0$ , such that

:: $|\alpha q - p - \beta| < \epsilon.$

Relation to tori

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

:T = R^N/Z^N,

the closure of the subgroup <P> generated by P will be finite, or some torus T′ contained in T. The original Kronecker's theorem (Leopold Kronecker, 1884) stated that the necessary condition for

:T′ = T,

which is that the numbers x_i together with 1 should be linearly independent over the rational numbers, is also sufficient. Here it is easy to see that if some linear combination of the x_i and 1 with non-zero rational number coefficients is zero, then the coefficients may be taken as integers, and a character χ of the group T other than the trivial character takes the value 1 on P. By Pontryagin duality we have T′ contained in the 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 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.

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}}

Category:Diophantine approximation

Category:Topological groups

Kronecker's theorem

Statement

Relation to tori

See also

References