The best diophantine approximation functions by continued fractions. (English) Zbl 0863.11042

If \(\xi\) is an irrational number with simple continued fraction expansion \(\xi = [a_0;a_1,a_2,\dots ,a_i,\dots ]\) and \(M_i= q_i^2|\xi -p_i/q_i|\) then lower, upper estimates (depending on \(R\) and \(r\)) are given for \(M_n\) if \(M_{n+1} < R\), \(M_{n-1} < r\) or \(M_{n+1} > R\), \(M_{n-1} > r\), respectively. An analytic expression for the estimate is presented.


11J04 Homogeneous approximation to one number
11A55 Continued fractions
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