Open AccessExplicit continued fractions with expected partial quotient growth
Author(s) -
Toshio Okano
Publication year - 2002
Publication title -
proceedings of the american mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.968
H-Index - 84
eISSN - 1088-6826
pISSN - 0002-9939
DOI - 10.1090/s0002-9939-02-06337-2
Subject(s) - algorithm , artificial intelligence , materials science , computer science
For 0 > x > 1 0>x>1 let [ 0 , a 1 ( x ) , a 2 ( x ) , … ] [0,a_1(x),a_2(x),\dots ] be the continued fraction expansion of x x . Write \[ L N ( x ) = max 1 ≤ n ≤ N a n ( x ) . L_N(x)=\max _{1\le n\le N}a_n(x). \] We construct some numbers x x ’s with \[ lim N → ∞ inf N − 1 L N ( x ) log log N = 1 / log 2. \lim _{N\to \infty }\inf N^{-1}L_N(x)\log \log N=1/\log 2. \]