bibkey: onishchik2020kronecker authors: A. L. Onishchik year: 2020 title: “Kronecker theorem” doi: null url: https://encyclopediaofmath.org/index.php?title=Kronecker_theorem&oldid=47528 claim: “The closure of a subgroup of a finite torus is determined by the integer characters that vanish on its generators; the single-generator case characterizes power-orbit closures.” strata_touched:
- D5/S3/Fourier/TorusOrbitClosure license: citation-only triage: anchor
Integer relations and torus subgroup closures
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Encyclopedia of Mathematics, “Kronecker theorem”, revision 47528, last edited June 5, 2020. The entry credits A. L. Onishchik as the originator of the original article. The year above identifies this revision, not the date of the classical theorem.
The opening theorem gives the simultaneous approximation criterion for vectors a_i and b in R^n: integer combinations of the a_i approximate b modulo Z^n exactly when every integer relation integral on all the a_i is also integral on b. The following paragraph, beginning “Kronecker’s theorem is a special case”, states the corresponding characterization of the closure of the generated subgroup of R^n/Z^n.
The entry attributes the original theorem to Kronecker in 1884 and cites Bourbaki’s General Topology and Pontryagin’s Topological Groups for the group formulation. Those books are bibliographic pointers; no theorem number or page locator from their bodies is asserted here.
Correspondence with the formal statement
Take one generator. Under the coordinatewise isomorphism from R/Z to Circle, x maps to exp(2 pi i x), and the integer character indexed by k maps a point g to the product of g_i raised to k_i. An additive integer relation becomes the multiplicative equality of this product with one. The source’s subgroup criterion therefore says that a point z belongs to the closure of the integer powers of g exactly when it satisfies every integer character relation satisfied by g.
In a compact group the closures of the integer and nonnegative power orbits coincide. The formal proof reuses this compact-group closure identity, so its natural-power formulation includes exponent zero. An arbitrary finite index type is a reindexing of the finite-dimensional torus; the empty index type additionally gives the singleton torus and empty products, both included in the formal statement.
No rational independence, algebraicity or density in the entire torus is assumed. Torsion generators and proper closed orbit subgroups are retained. The formal proof uses Haar averaging, density of torus characters and continuous separation of compact cosets. This proof construction does not assert originality of the classical criterion.
The note attests the topological characterization only. It supplies no algorithm for computing a finite relation basis and no algebraic optimization or semidefinite-programming assertion.