Kronecker's theorem on diophantine approximation

In mathematics, Kronecker's theorem is a theorem about diophantine approximation, introduced by Leopold Kronecker (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.

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.

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 and integer vectors.

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