On a family of singular measures related to Minkowski's?(x) function

In the present paper we are investigating a certain point measure of a distribution function arising in a paper by Grabner et al. [Combinatorica 22 (2002) 245–267]. This distribution function is defined by means of the subtractive Euclidean algorithm and bears a striking resemblance to the singular?(x)-function of H. Minkowski. Beyond it, we will also consider a whole family of distribution functions arising in a natural way from the above ones. Nevertheless we will prove that all of the corresponding measures of the mentioned functions are mutually singular by using dynamical systems and the ergodic theorem.