Kronecker's theorem

Template:Short description Template:For In mathematics, Kronecker's theorem is a theorem about diophantine approximation, introduced by Template:Harvs.

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 <math>\alpha_i=(\alpha_{i 1},\dots,\alpha_{i n})\in\mathbb{R}^n, i=1,\dots,m </math> and <math>\beta=(\beta_1,\dots,\beta_n)\in \mathbb{R}^n</math> , the condition:

<math>\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</math>

holds if and only if for any <math>r_1,\dots,r_n\in\mathbb{Z},\ i=1,\dots,m</math> with

<math>\sum^n_{j=1}\alpha_{ij}r_j\in\mathbb{Z}, \ \ i=1,\dots,m\ ,</math>

the number <math>\sum^n_{j=1}\beta_jr_j</math> is also an integer.

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

For the case of a <math>m=1</math> and <math>n=1</math>, Kronecker's theorem can be stated as follows.<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref> For any <math>\alpha, \beta, \epsilon \in \mathbb{R}</math> with <math>\alpha</math> irrational and <math>\epsilon > 0</math> there exist integers <math>p</math> and <math>q</math> with <math>q>0</math>, such that

<math>|\alpha q - p - \beta| < \epsilon.</math>

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.

See also

• Weyl's criterion
• Dirichlet's approximation theorem

References

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