Kronecker's theorem

Kronecker's theorem is a result in diophantine approximations applying to several real numbers x_i, for 1 ≤ i ≤ n, that generalises Dirichlet's approximation theorem to multiple variables.

The classical Kronecker approximation theorem is formulated as follows.

Given real n-tuples ${\displaystyle \alpha _{i}=(\alpha _{i1},\cdots ,\alpha _{in})\in \mathbb {R} ^{n},i=1,\cdots ,m}$ and ${\displaystyle \beta =(\beta _{1},\cdots ,\beta _{n})\in \mathbb {R} ^{n}}$ , the condition:

${\displaystyle \forall \epsilon >0\,\exists q_{i},p_{j}\in \mathbb {Z}$ :{\biggl |}\sum _{i=1}^{m}q_{i}\alpha _{ij}-p_{j}-\beta _{j}{\biggr |}<\epsilon ,1\leq j\leq n}

holds if and only if for any ${\displaystyle r_{1},\dots ,r_{n}\in \mathbb {Z} ,\ i=1,\dots ,m}$ with

${\displaystyle \sum _{j=1}^{n}\alpha _{ij}r_{j}\in \mathbb {Z} ,\ \ i=1,\dots ,m\ ,}$

the number ${\displaystyle \sum _{j=1}^{n}\beta _{j}r_{j}}$ is also an integer.

In plainer language the first condition states that the tuple ${\displaystyle \beta =(\beta _{1},\ldots ,\beta _{n})}$ can be approximated arbitrarily well by linear combinations of the ${\displaystyle \alpha _{i}}$s (with integer coefficients) and integer vectors.

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

${\displaystyle |\alpha q-p-\beta |<\epsilon .}$

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.

• Weyl's criterion
• Dirichlet's approximation theorem

• Kronecker, L. (1884), "Näherungsweise ganzzahlige Auflösung linearer Gleichungen", Berl. Ber.: 1179–1193, 1271–1299
• Onishchik, A.L. (2001) [1994], "Kronecker's theorem", Encyclopedia of Mathematics, EMS Press