Dimension of countable intersections of some sets arising in expansions in non-integer bases
We consider expansions of real numbers in non-integer bases. These expansions are generated by $\beta $-shifts. We prove that some sets arising in metric number theory have the countable intersection property. This allows us to consider sets of reals that have common properties in a countable number of different (non-integer) bases. Some of the results are new even for integer bases.