Quantitative metric theory of continued fractions

Abstract

Quantitative versions of the central results of the metric theory of continued fractions were given primarily by C. De Vroedt. In this paper we give improvements of the bounds involved . For a real number x, let $$x \ = \ c_{0} + \frac{1}{\displaystyle c_{1} + \frac{1}{\displaystyle c_{2} + \frac{1}{\displaystyle c_{3} + \frac{1}{\displaystyle c_{4} +_{\ddots}}}}}. $$ A sample result we prove is that given 𝜖 > 0, $$\left (c_{1} (x) {\cdots} c_{n}(x) \right )^{\frac{1}{n}} = { \prod}_{k=1}^{\infty}\left (1+ \frac{1}{k(k+2)} \right )^{\frac{\log k }{ \log 2}} + o \left (n^{-\frac{1}{ 2}}(\log n )^{\frac{3}{ 2}} (\log \log n)^{\frac{1}{2}+\epsilon} \right ) $$ almost everywhere with respect to the Lebesgue measure.