Quantitative metric theory of continued fractions

Proceedings Mathematical Sciences

P Lertchoosakul⁴  J Hančl³  A Haddley¹  R Nair² 

[1] Mathematical Sciences, The University of Liverpool, Peach Street, Liverpool L ZL, United Kingdom$$;$$;Department of Mathematics and Centre for Excellence ITInnovation, Division of UO, Institute for Research and Applications of Fuzzy Modeling, University of Ostrava, 0. dubna , 0 0 Ostrava , Czech Republic$$;Institute of Mathematics, Polish Academy of Sciences, ul Åšniadechich , 00- Warsaw, Poland$$

关键词: Continued fractions;    ergodic averages;    metric theory of numbers.;   

DOI  :  

学科分类：数学（综合）

来源: Indian Academy of Sciences

【 摘 要 】

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

【 授权许可】

Unknown