J Austral Math Soc Ser A 51 pp324--330, 1991.

Diophantine Approximation by Continued Fractions

Jingcheng Tong

Abstract

Let x be an irrational number with simple continued fraction expansion

x = [a0 ; a1, ..., ai, ...],

p_i/q_i be its ith convergent. Let M_i = [a_i+1 ; a_i, ..., a₁] + [0 ; a_i+2, a_i+3, ...]. In this paper we prove that M_n-1 < r and M_n < R imply M_n+1 > 1/(r^-1 + a_n+1Ö(1 - 4/(rR)) - a_n+1²R^-1), which generalizes a previous result of the author.

1980 AMS Subject Classification (1985 Revision): 11J04, 11A55

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Authors

Jingcheng Tong

Department of Mathematics and Statistics, University of North Florida, Jacksonville, Florida 32216, U.S.A.