I already pointed this out privately, but since it continues to be discussed I'll repeat and expand on my answer. Yes, expansions in alternate bases have been discussed in the literature, and in fact, enormously so. If you search the on-line Math Reviews (mathscinet) for "beta-expansion" you will get something on the order of 60 references. The connection with Pisot numbers has, in particular, been studied extensively by Bertrand-Mathis, Frougny, Boyd, Solomyak, and others. (You should also search for the term "theta-development", which has sometimes been used by French people who don't know any better.) On the other hand, expansions in base-k with alternate digits have also been intensively studied, and there are indeed some results on what sets of digits are necessary and sufficient to produce expansions for all numbers. For example, see the paper of Matula, Basic digit sets for radix representation, J. ACM 29 (1982), 1131-1143. There is also a paper by Odlyzko. Prof. Thurston has already mentioned the theory of "automatic groups", but I'd also like to advertise the area I work in: the area of "automatic sequences". In fact, Allouche and I have finished a 570-page book on the subject, which will be published soon by Cambridge University Press. This book has a chapter on representations in base k and other sorts of representations, including a presentation of Matula's result. Jeffrey Shallit