Do there exist numbers with non-integer irrationality measure?

Question

Is it possible to construct a number (by way of an infinite series or a continued fraction say) having any, possibly non-integer, irrationality measure $>2$ ? It is known that this can be done for integer irrationality measures: the Champernowne constant in base b has irrationality measure equal to b, for example. There are also some seemingly good candidates like the family of series $ \sum_{k=0}^{\infty} 10^{-\lfloor b^k \rfloor} $ for $ b >1$, but they have irrationality measure at least b, not exactly b. Any help would be greatly appreciated.

Answer 1

According to and using the notation of http://mathworld.wolfram.com/IrrationalityMeasure.html , all you need is $\limsup\frac{\ln a_{n+1}}{\ln q_n}=\mu-2$. As $q_n\to \infty$ is guaranteed, you can recursively let $a_{n+1}=\lceil q_n^{\mu-2}\rceil$.