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.
The classical Kronecker approximation theorem is formulated as follows.
- Given real n-tuples and , the condition:
- holds if and only if for any with
- the number is also an integer.
In plainer language the first condition states that the tuple can be approximated arbitrarily well by linear combinations of the s (with integer coefficients) and integer vectors.
For the case of a and , Kronecker's Approximation Theorem can be stated as follows. For any with irrational and there exist integers and with , such that
Relation to tori
- T = RN/ZN,
- T′ = T,
which is that the numbers xi 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 xi 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.
- Kronecker, L. (1884), "Näherungsweise ganzzahlige Auflösung linearer Gleichungen", Berl. Ber.: 1179–1193, 1271–1299
- Onishchik, A.L. (2001) , "Kronecker's theorem", Encyclopedia of Mathematics, EMS Press
- "Kronecker's Approximation Theorem". Wolfram Mathworld. Retrieved 2019-10-26.