Consider the expanding map given by multiplication by a non-integer modulo 1. The typical behavior for a point under this map is that the orbit is dense and that ergodic averages converge. However, the set of exceptional points, not having such properties, has full Hausdorff dimension and even has large intersection properties. I will show how this can be proven. Central for the argument is the β-shift, the symbolic space generated by the map. The case when this shift is not of finite type is tricky and is handled by an approximation argument.