One way to obtain explicit examples, which combines the ideas of (weak forms of) randomness and base 3 expansions, is to use the fact that normality in a given base is preserved under rational addition, which was proved by D. D. Wall in his 1949 Berkeley PhD Dissertation. (I'm relying on D. Doty, J. H. Lutz, and S.Nandakumar [*Finite-state dimension and real arithmetic*, Information and Computation 205(11):1640-1651, 2007] for this reference.) Here a number $r$ is *normal in base $b$* if for any finite nonempty string $\sigma$ drawn from the alphabet $\{0,...,b-1\}$, the limiting frequency of the appearances of $\sigma$ as a substring of the base $b$ expansion of $r$ is $b^{-|\sigma|}$. Since elements of $C$ are not normal in base $3$, any number $r$ that *is* normal in base $3$ has the desired property. Examples of such numbers can be found at http://en.wikipedia.org/wiki/Normal_number, for instance.