Let 1 < q < 2 be non-integer; then, as is well known, any x from [0,1] can be represented as a series in decreasing powers of q with `digits' 0 and 1 (one can apply the greedy algorithm, say). It is known that for any fixed q a generic x has a continuum of such expansions.
In my talk, however, I will concentrate on the exceptional set of x with a `small' number of such expansions. A dynamical interpretation of this arithmetic model will be presented, as well as a (perhaps, surprising) connection with the theory of sums of Cantor sets.
Time permitting, I will also explain how to generalize this model to the case of higher dimensions (which involves the language of iterated function systems).