THE BEST APPROXIMATION FUNCTION TO IRRATIONAL NUMBERS

[摘要] Let zeta be an irrational number with simple continued fraction expansion zeta = [a0; a1,a2, ..., a(i),...]. Let the ith convergent p(i)/q(i) = [a0; a1, a2...,a(i)]. Let mu = \[0; a(n) + 2, a(n) + 3, ...] - [0; a(n)-1,..., a1]\. In this note, we prove that among three consecutive convergents p(i)/q(i) (i=n-1, n, n + 1), at least one satisfies \zeta - p(i)/q(i)< 1/(square root (a(n) + 1 + mum)2 + 4 q(i)2), and at least one satisfies \zeta - p(i)/q(i) > 1/(square root (a(n)+1)- mu)2 + 4 q(i)2). The results are best possible. (C) 1994 Academic Press, Inc.

[发布日期] 1994-10-01