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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 Khinchin's constant ： ウィキペディア英語版 Khinchin's constant In number theory, Aleksandr Yakovlevich Khinchin proved that for almost all real numbers ''x'', coefficients ''a''''i'' of the continued fraction expansion of ''x'' have a finite geometric mean that is independent of the value of ''x'' and is known as Khinchin's constant. That is, for :$x\; =\; a\_0+\backslash cfrac\}\}\}\backslash ;$ it is almost always true that :$\backslash lim\_\; \backslash left(\; a\_1\; a\_2\; ...\; a\_n\; \backslash right)\; ^\; =$ K_0 where $K\_0$ is Khinchin's constant :$K\_0\; =$ \prod_^\infty ^ \approx 2.6854520010\dots (with $\backslash prod$ denoting the product over all sequence terms). But although almost all numbers satisfy this property, it has not been proven for ''any'' real number ''not'' specifically constructed for the purpose. Among the numbers ''x'' whose continued fraction expansions are known ''not'' to have this property are rational numbers, roots of quadratic equations (including the square roots of integers and the golden ratio Φ), and the base of the natural logarithm ''e''. Khinchin is sometimes spelled Khintchine (the French transliteration of Russian Хи́нчин) in older mathematical literature. ==Sketch of proof== The proof presented here was arranged by and is much simpler than Khinchin's original proof which did not use ergodic theory. Since the first coefficient ''a''0 of the continued fraction of ''x'' plays no role in Khinchin's theorem and since the rational numbers have Lebesgue measure zero, we are reduced to the study of irrational numbers in the unit interval, i.e., those in $\backslash scriptstyle\; I=()\backslash setminus\backslash mathbb$. These numbers are in bijection with infinite continued fractions of the form (), which we simply write (), where ''a''1, ''a''2, ... are positive integers. Define a transformation ''T'':''I'' → ''I'' by :$T(())=().\backslash ,$ The transformation ''T'' is called the Gauss–Kuzmin–Wirsing operator. For every Borel subset ''E'' of ''I'', we also define the Gauss–Kuzmin measure of ''E'' :$\backslash mu(E)=\backslash frac\backslash int\_E\backslash frac.$ Then ''μ'' is a probability measure on the ''σ''-algebra of Borel subsets of ''I''. The measure ''μ'' is equivalent to the Lebesgue measure on ''I'', but it has the additional property that the transformation ''T'' preserves the measure ''μ''. Moreover, it can be proved that ''T'' is an ergodic transformation of the measurable space ''I'' endowed with the probability measure ''μ'' (this is the hard part of the proof). The ergodic theorem then says that for any ''μ''-integrable function ''f'' on ''I'', the average value of $f\; \backslash left(\; T^k\; x\; \backslash right)$ is the same for almost all $x$: :$\backslash lim\_\; \backslash frac\; 1n\backslash sum\_^(f\backslash circ\; T^k)(x)=\backslash int\_I\; f\; d\backslash mu\backslash quad\backslash text\backslash mu\backslash textx\backslash in\; I.$ Applying this to the function defined by ''f''(()) = log(''a''1), we obtain that :$\backslash lim\_\backslash frac\; 1n\backslash sum\_^\backslash log(a\_k)=\backslash int\_I\; f\; \backslash ,\; d\backslash mu\; =\; \backslash sum\_^\backslash infty\backslash log(r)\backslash frac\backslash bigr)\}$ for almost all () in ''I'' as ''n'' → ∞. Taking the exponential on both sides, we obtain to the left the geometric mean of the first ''n'' coefficients of the continued fraction, and to the right Khinchin's constant. 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「Khinchin's constant」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.