Diophantine approximation of a single irrational number. (English) Zbl 0714.11036

For an irrational number x with simple continued fraction expansion \(x=[0;a_ 1,a_ 2,...]\) and (reduced) convergents \(p_ n/q_ n=[0;a_ 1,...,a_ n]\) let \(Q_ n=[a_ n,...,a_ 1]\), \(P_ n=[a_{n+2},a_{n+3},...]\). Rational approximation of x rests on the identity \(M_ n^{-1}=(-1)^ nq_ n(q_ nx-p_ n)\), where \(M_ n=a_{n+1}+P_ n^{-1}+Q_ n^{-1}\). Put \(f(Q,M)=Q+(M-Q^{-1})^{- 1}\). The following relations are evident: \(M_{n-1}=f(Q_ n,M_ n)\), \(M_{n+1}=f(P_ n,M_ n)\). The simple observation that f is an increasing function of Q, provided \(MQ>2\), leads to the following surprisingly useful implications, setting a foundation for the theory of diophantine approximation (symmetric or one-sided) of a single variable. Theorem: Let \(r,s\in {\mathbb{R}}_+\), \(r>a_{n+1}\), \(r-a_{n+1}-s^{- 1}\neq 0\). Put \(g_ n=(r-a_{n+1}-s^{-1})^{-1}+(a_{n+1}+s^{- 1})^{-1}\). Then \(M_ n\leq r\), \(P_ n<s\) imply \(M_{n-1}>g_ n\); \(M_ n\leq r\), \(Q_ n<s\) imply \(M_{n+1}>g_ n\); \(M_ n\geq r\), \(P_ n>s\) imply \(M_{n-1}<g_ n\); \(M_ n\geq r\), \(Q_ n>s\) imply \(M_{n+1}<g_ n\). Appropriate choice of r and s yields numerous known results widely spread in the literature.
Reviewer: G.Ramharter


11J04 Homogeneous approximation to one number
11J70 Continued fractions and generalizations
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