Theorem

Kronecker's Approximation Theorem

Number Theory

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

Statement

For 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

Proof Year

1884 1

Classification

Statement Form

Existence Theorem 1

Statement Form

Inequality 1

Statement Form

Characterization Theorem 1

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).

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Kronecker's Approximation Theorem