Segre's theorem on asymmetric diophantine approximation
The following theorem is proved: Let \(\xi\) be a real irrational number, \(p_i/q_i\) \((i=1,2,\ldots)\) the convergents of its regular continued fraction expansion. Then we have for any even \(n\ge 2\) \[ -(a^2_{n+1}+4\tau)^{-} < (-1)^{n+1}q^2_1(\xi - p_i/q_i) < \tau(a^2_{n+1}+4\tau)^{-} \] at least once for \(i=n-1,n,n+1\); here \(a_{i+1}\) is the \(n+1\)st partial quotient of the continued fraction expansion of \(\xi\) and \(\tau\) an arbitrary positive number. As corollaries of the above theorem the proof of a conjecture of LeVeque and a proof of the well-known fact are given that any irrational number not equivalent to \((5^{1/2}-1)/2\) admits a rational approximation \(| \xi - p/q| < (8^{1/2}q^ 2)^{-1}\).
scientific article; zbMATH DE number 4051713