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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Dirichlet's approximation theorem ： ウィキペディア英語版 Dirichlet's approximation theorem In number theory, Dirichlet's theorem on Diophantine approximation, also called Dirichlet's approximation theorem, states that for any real number α and any positive integer ''N'', there exists integers ''p'' and ''q'' such that 1 ≤ ''q'' ≤ ''N'' and :$\backslash left\; |\; q\; \backslash alpha\; -p\; \backslash right\; |\; \backslash le\; \backslash frac$ This is a fundamental result in Diophantine approximation, showing that any real number has a sequence of good rational approximations: in fact an immediate consequence is that for a given irrational α, the inequality : is satisfied by infinitely many integers ''p'' and ''q''. This corollary also shows that the Thue–Siegel–Roth theorem, a result in the other direction, provides essentially the tightest possible bound, in the sense that the limits on rational approximation of algebraic numbers cannot be improved by lowering the exponent 2 + ε beyond 2. ==Simultaneous Version== The simultaneous version of the Dirichlet's approximation theorem states that given real numbers $\backslash alpha\_1,\; ...,\; \backslash alpha\_d$ and a natural number $N$ then there are integers $p\_1,\; ...,\; p\_d,\; q\backslash in\backslash Z,1\backslash le\; q\backslash leq\; N$ such that $\backslash left|\backslash alpha\_i-\backslash fracq\backslash right|\backslash le\backslash frac1\{qN^\{1/d\}\}.$ 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Dirichlet's approximation theorem」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.