Dirichlet box principle

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A theorem according to which any sample of $ n $ sets containing in total more than $ n $ elements comprises at least one set with at least two elements. Dirichlet's box principle can be formulated in a most popular manner as follows: If $ n $" boxes" contain $ n + 1 $" objects" , then at least one "box" contains at least two "objects" . The principle is frequently used in the theory of Diophantine approximations and in the theory of transcendental numbers to prove that a system of linear inequalities can be solved in integers (cf. Dirichlet theorem in the theory of Diophantine approximations).

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Dirichlet box principle. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by V.G. Sprindzhuk (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article