Let $C$ be the standard Cantor middle-third set. As a consequence of the Baire category theorem, there are numbers $r$ such that $C+r$ consists solely of irrational numbers, see [here][1]. 

> What would be an explicit example of a number $r$ with this property?

Short of an explicit example, are there any references addressing this question?


  [1]: http://math.stackexchange.com/q/381690/462