## 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

### MSC:

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

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