  We consider expanding maps such that the unit interval can be represented as
a full symbolic shift space with bounded distortion. There are already theorems
about the Hausdorff dimension for sets defined by the set of accumulation
points for the frequencies of words in one symbolic space at a time. It is
shown in this text that the dimension is preserved when sets defined using
different maps are intersected. More precisely, it is proven that the dimension
of any countable intersection of sets defined by their sets of accumulation for
frequencies of words in different expansions, has dimension equal to the
infimum of the dimensions of the sets that are intersected. As a consequence,
the set of numbers for which the frequencies do not exist has full dimension
even after countable intersections. We prove these results also for a dense set
of $\beta$-shifts.
