An illustration for the answer by Gjergji Zaimi: the mysterious sequence $\epsilon_1,\epsilon_2,...,\epsilon_{5000}$
And here, in response to the comment by André Henriques, is the plot of lengths of gaps between consecutive points in the set $\{\epsilon_k\mid10000<k<20000\}$ rearranged in the increasing order. Some Cantor-set-like structure seems to be present indeed.
