Kronecker's theorem

In mathematics, Kronecker's theorem is a result in diophantine approximation applying to several real numbers xi, for 1 ≤ i ≤ N, which generalises the equidistribution theorem, the fact that an infinite cyclic subgroup of the unit circle group is a dense subset. 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.

Related Topics:
Mathematics - Diophantine approximation - Real number - Equidistribution theorem - Infinite cyclic - Unit circle - Dense

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In the case of N numbers, taken as a single N-tuple and point P of the torus

Related Topics:
Tuple - Torus

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:T = RN/ZN,

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the closure of the subgroup 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

Related Topics:
Closure - Leopold Kronecker - Necessary condition

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:T′ = T,

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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.

Related Topics:
Linearly independent - Rational number - Sufficient - Linear combination - Character - Trivial character - Pontryagin duality - Kernel

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In fact a thorough use of Pontryagin duality here shows that the whole Kronecker theorem describes the closure of as the intersection of the kernels of the χ with

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:χ(P) = 1.

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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.

Related Topics:
Antitone - Galois connection

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The theorem leaves open the question of how well (uniformly) the multiples mP of P fill up the closure.

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See also: Kronecker set

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