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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



On the metrical theory of continued fractions

Author: R. Nair
Journal: Proc. Amer. Math. Soc. 120 (1994), 1041-1046
MSC: Primary 11K50; Secondary 28D05
MathSciNet review: 1176073
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Abstract: Suppose $ {b_k}$ denotes either $ \phi (k)$ or $ \phi ({p_k})\;(k = 1,2, \ldots )$ where the polynomial $ \phi $ maps $ \mathbb{N}$ to $ \mathbb{N}$ and $ {p_k}$ denotes the $ k$th rational prime. Suppose $ ({c_k}(x))_{k = 1}^\infty $ denotes the sequences of partial quotients of the continued function expansion of the real number $ x$. Then for certain functions $ F:{\mathbb{R}_{ \geqslant 0}} \to \mathbb{R}$ we show that

$\displaystyle \mathop {\lim }\limits_{N \to \infty } {F^{ - 1}}\left[ {\frac{{F... ...ft[ {\frac{1} {{(\log 2)}}\int_0^1 {\frac{{F({c_1}(x))}} {{1 + x}}dx} } \right]$

almost everywhere with respect to Lebesgue measure. This result with $ {b_k} = k$ is classical and due to Ryll-Nardzewski.

References [Enhancements On Off] (What's this?)

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