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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 Duffin–Schaeffer conjecture ： ウィキペディア英語版 Duffin–Schaeffer conjecture The Duffin–Schaeffer conjecture is an important conjecture in metric number theory proposed by R. J. Duffin and A. C. Schaeffer in 1941. It states that if $f\; :\; \backslash mathbb\; \backslash rightarrow\; \backslash mathbb^+$ is a real-valued function taking on positive values, then for almost all $\backslash alpha$ (with respect to Lebesgue measure), the inequality : has infinitely many solutions in co-prime integers $p,q$ with $q\; >\; 0$ if and only if the sum : $\backslash sum\_^\backslash infty\; f(q)\; \backslash frac\; =\; \backslash infty.$ Here $\backslash varphi(q)$ is the Euler totient function. The full conjecture remains unsolved. However, a higher-dimensional analogue of this conjecture has been resolved.〔〔Harman (2002) p.69〕 ==Progress== The implication from the existence of the rational approximations to the divergence of the series follows from the Borel–Cantelli lemma.〔Harman (2002) p.68〕 The converse implication is the crux of the conjecture.〔 There have been many partial results of the Duffin–Schaeffer conjecture established to date. Paul Erdős established in 1970 that the conjecture holds if there exists a constant $c\; >\; 0$ such that for every integer $n$ we have either $f(n)\; =\; c/n$ or $f(n)\; =\; 0$.〔Harman (1998) p.27〕 This was strengthened by Jeffrey Vaaler in 1978 to the case $f(n)\; =\; O(n^)$.〔http://www.math.osu.edu/files/duffin-schaeffer%20conjecture.pdf〕〔Harman (1998) p.28〕 More recently, this was strengthened to the conjecture being true whenever there exists some $\backslash epsilon\; >\; 0$ such that the series $\backslash sum\_^\backslash infty\; \backslash left(\backslash frac\backslash right)^\; \backslash varphi(n)\; =\; \backslash infty$. This was done by Haynes, Pollington, and Velani.〔A. Haynes, A. Pollington, and S. Velani, ''The Duffin-Schaeffer Conjecture with extra divergence'', arXiv, (2009), http://arxiv.org/abs/0811.1234〕 In 2006, Beresnevich and Velani proved that a Hausdorff measure analogue of the Duffin–Schaeffer conjecture is equivalent to the original Duffin–Schaeffer conjecture, which is a priori weaker. This result is published in the ''Annals of Mathematics''. 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「Duffin–Schaeffer conjecture」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.