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.
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 α i = ( α i 1 , … , α i n ) ∈ R n , i = 1 , … , m {\displaystyle \alpha _{i}=(\alpha _{i1},\dots ,\alpha _{in})\in \mathbb {R} ^{n},i=1,\dots ,m} and β = ( β 1 , … , β n ) ∈ R n {\displaystyle \beta =(\beta _{1},\dots ,\beta _{n})\in \mathbb {R} ^{n}} , the condition:
∀ ϵ > 0 ∃ q i , p j ∈ Z : | ∑ i = 1 m q i α i j − p j − β j | < ϵ , 1 ≤ j ≤ n {\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 r 1 , … , r n ∈ Z , i = 1 , … , m {\displaystyle r_{1},\dots ,r_{n}\in \mathbb {Z} ,\ i=1,\dots ,m} with
∑ j = 1 n α i j r j ∈ Z , i = 1 , … , m , {\displaystyle \sum _{j=1}^{n}\alpha _{ij}r_{j}\in \mathbb {Z} ,\ \ i=1,\dots ,m\ ,}
the number ∑ j = 1 n β j r j {\displaystyle \sum _{j=1}^{n}\beta _{j}r_{j}} is also an integer. In plainer language, the first condition states that the tuple β = ( β 1 , … , β n ) {\displaystyle \beta =(\beta _{1},\ldots ,\beta _{n})} can be approximated arbitrarily well by linear combinations of the α i {\displaystyle \alpha _{i}} s (with integer coefficients) and integer vectors. For the case of a m = 1 {\displaystyle m=1} and n = 1 {\displaystyle n=1} , Kronecker's theorem can be stated as follows. For any α , β , ϵ ∈ R {\displaystyle \alpha ,\beta ,\epsilon \in \mathbb {R} } with α {\displaystyle \alpha } irrational and ϵ > 0 {\displaystyle \epsilon >0} there exist integers p {\displaystyle p} and q {\displaystyle q} with q > 0 {\displaystyle q>0} , such that
| α q − p − β | < ϵ . {\displaystyle |\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 = RN/ZN, 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
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