Kronecker's Approximation Theorem states that if a collection of real numbers together with the number one is linearly independent over the rational numbers, then the sequence of their integer multiples, taken modulo one, comes arbitrarily close to every point of the unit cube, so no proper closed subgroup of the torus can contain the whole orbit. Named for Leopold Kronecker, it is a classical result of Diophantine approximation describing the density of irrational rotations.
Facts
StatementFor irrational alpha and any epsilon > 0, there exist integers p and q with q > 0 such that |alpha q - p - beta| < epsilon; the general form characterizes simultaneous approximation by integer relations. 1 Classification
Statement Form Statement Form Statement FormCharacterization Theorem 1 Connections
Has Statement Form
Entity-backed identity for the statement-form enum value this theorem already carries, resolved to a mathematics concept by an explicit value-to-entity map (phase 3 bucket conversion, docs\design_entity_backed_browse_buckets_20260928.md). The statement-form fact itself stays on the theorem unchanged.
Entity-backed identity for the statement-form enum value this theorem already carries, resolved to a mathematics concept by an explicit value-to-entity map (phase 3 bucket conversion, docs\design_entity_backed_browse_buckets_20260928.md). The statement-form fact itself stays on the theorem unchanged.
Entity-backed identity for the statement-form enum value this theorem already carries, resolved to a mathematics concept by an explicit value-to-entity map (phase 3 bucket conversion, docs\design_entity_backed_browse_buckets_20260928.md). The statement-form fact itself stays on the theorem unchanged.
Proved By
Source Kronecker's theorem (Wikipedia)
Sources
1. Kronecker's theorem (Wikipedia)
Statement
For the simpler case, when α is irrational and ϵ > 0, there exist integers p and q where q > 0, such that | α q − p − β | < ϵ.
Historical Context
introduced by Leopold Kronecker (1884).
Proved By: Leopold Kronecker, Lead paragraph
In mathematics, Kronecker's theorem is a theorem about diophantine approximation, introduced by Leopold Kronecker (1884).
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