Be aware that http://en.wikipedia.org/wiki/Thue%E2%80%93Morse_sequence" is not \frac{(1+\beta)}{100} as it goes on .01101001100101101001011001101001... so even if you found a proof for it, that wouldn't apply to your number anyway.
Unfortunately not. As I said, this was a test sheet and I don't have access to the relevant textbook. Anyway I'd say it's obvious now that this is not standard notation, I'll check it on the book itself when it becomes available.
Edit: For all the curious here, I managed to obtain the...
Thank you for your quick answer. I doubt it to be anything more complicated than very basic set theory. Unfortunately, I don't know the context myself, all I have is a simple test asking to answer true or false to several questions:
if l belogs to A* then l+1 does not belong to A*;
if l...