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Rational approximations to the Rogers-Ramanujan continued fraction. (English) Zbl 0655.10031

Let \(f(\alpha,x)=1+\alpha x/(1+\alpha x^ 2/(1+\alpha x^ 3/(1+...]\) be the Rogers-Ramanujan continued fraction and let a, b, c, d be non-zero integers with \(| d| >c^ 2\). The following is proved: \(z=f(a/b,c/d)\) is an irrational number, and there exists a positive constant C depending on these integers only such that the inequality \(| z-p/q| \leq C/q^{2+2A+B\epsilon (q)}\) has only finitely many solutions in (reduced) fractions p/q. Here \(A=2(\log | c|)/\log | d/c^ 2|\), \(\epsilon (q)=(\log q)^{-1/2},\) and B is an explicit constant depending on a, b, c, d. As \(A=0\) if \(c=1\), one obtains as a corollary that f(a/b,1/d) does not admit an approximation with \(C/q^{2+B\epsilon (q)}\). This improves upon a result by C. F. Osgood [J. Number Theory 3, 159-177 (1971; Zbl 0218.10051)].
The above approximation theorem is best possible in the sense that for positive integers a, b, d with a, b relatively prime, a dividing d, \(d\geq 2\), there is an (explicit) constant C’ such that f(a/b,1/d) admits an approximation with \(D/q^{2+E\epsilon (q)}\) \((E=(\log d)^{1/2})\), whenever \(D>C'\), but not if \(D<C'\).
Reviewer: G.Ramharter

MSC:

11J70 Continued fractions and generalizations
11J04 Homogeneous approximation to one number
11A55 Continued fractions

Citations:

Zbl 0218.10051
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