Does there exist a dense subset $\ X\ $ of $\ \mathbb{R}\ $ and a real number $\ a\neq 0\ $ such that $\{\ x+a:\ x\in X\ \} =\mathbb{R}\setminus X\ ?$ Does there exist a dense subset $\ X\ $ of $\ \mathbb{R},\ $ and a real number $\ a\neq 0\ $ such that $\ \{\ x+a:\ x\in X\ \} = \mathbb{R}\setminus X\ ?$
Clearly our set $\ X\ $ must be totally disconnected and uncountable on every interval in order for $\ X\ $ to exist.
But I'm actually not sure on the answer to this question, and find it hard to give even a heuristic argument as to why the answer should be negative or affirmative. Maybe it has something to do with the Baire Category Theorem?

$$X=\{ x \in \mathbb R : \lfloor x\rfloor\text{ is odd}\}\mathbin{\triangle}  \mathbb Q = \bigcup_{n\in\mathbb Z}\,\Bigl([2n,2n+1)\cap\mathbb Q\Bigr) \cup \Bigl([2n+1,2n+2) \setminus \mathbb Q\Bigr) $$
This works for $a=1$. Scale it by your favorite $a$ if you want a different one.
(This $X$ is countable on every other unit interval, by the way).
