Approximation of the values of certain functions (Q1823976)

Let R be the ring of integers in the rational or in some imaginary quadratic field. Let \(C_ 0,C_ 1,..\). be a sequence of units in R. (Thus, except for two cases, \(C_ j\in \{1,-1\}.)\) Put \(f(z)=\sum^{\infty}_{j=0}C_ j z^{2^ j}\). Theorem: For arbitrary b, p, q in R such that \(| b| >1\), the inequality \[ | f(\frac{1}{b})-\frac{p}{q}| \quad >\quad \frac{1}{4}\cdot \frac{(| b| -1)^ 2}{| b|^ 4}\cdot \frac{1}{| q|^ 2} \] holds. \textit{K. Mahler} [Math. Ann. 101, 342-366 (1929; JFM 55.0115.01)] initiated the study of arithmetical properties of numbers like f(\(\alpha)\). In the rational case, the continued fractions of f(1/b) were studied by \textit{J. Shallit} [J. Number Theory 11, 209-217 (1979; Zbl 0404.10003)] and the reviewer [Monatsh. Math. 89, 95-100 (1980; Zbl 0419.10010)].