Irrationality exponents of semi-regular continued fractions (Q6611870)

Consider an irrational number \(\alpha\) with a regular continued fraction \([b_0, b_1, b_2, \ldots]\) and let \(p_n/q_n\) be its convergents. As it is known the irrationality exponent of \(\alpha\) is given by the \N\[ \N\mu(a)=2+ \lim\sup_{n\to \infty} (\log b_{n+1}/\log q_n).\N\] \NThe authors show that this formula holds for semi-regular continued fractions (where the operations \(+\) involved in the continued fraction expression are partially replaced by \(-\)) satisfying certain conditions. For instance it holds for the nearest integer and singular continued fractions and does not hold for the negative and farthest integer continued fractions. The authors illustrate the paper with several examples of exact computation of irrationality exponents of semi-regular continued fractions.