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 e?‘￥, let$$x=c_0+dfrac{1}{c_1+dfrac{1}{c_2+dfrac{1}{c_3+dfrac{1}{c_4+_ddots}}}}.$$A sample result we prove is that given $epsilon 0$,$$(c_1(x)cdots c_n(x))^{frac{1}{n}}=prod^infty_{k=1}left( 1+frac{1}{k(k+2)} ight)^{frac{log , k}{log , 2}}+oleft(n^{-frac{1}{2}}(log , n)^{frac{3}{2}}(log , log , n)^{frac{1}{2}+epsilon} ight)$$
展开▼
