I am familiar with the work of Helmut Cajar, but his book is thirty years old and it's clear that there has been substantial progress since then. I have been spending a lot of time looking through mathscinet citations and am honestly a bit overwhelmed. Also, I'm not just interested in sets defined through restrictions on digits of their decimals expansions, but also similar results pertaining to continued fraction expansion, Cantor series expansions, the Luroth series expansion, Engel series expansion, etc.

I would guess you'll find many recent pointers in the publication list of Lars Olsen, and in that of Godofredo Iommi, for example: 

