CN-TST 2017 Problem 12

Let $M$ be a subset of $\mathbb{R}$ such that the following conditions are satisfied:

a) For any $x \in M, n \in \mathbb{Z}$ , one has that $x+n \in \mathbb{M}$ . b) For any $x \in M$ , one has that $-x \in M$ . c) Both $M$ and $\mathbb{R}$ \ $M$ contain an interval of length larger than $0$ .

For any real $x$ , let $M(x) = \{ n \in \mathbb{Z}^{+} | nx \in M \}$ . Show that if $\alpha,\beta$ are reals such that $M(\alpha) = M(\beta)$ , then we must have one of $\alpha + \beta$ and $\alpha - \beta$ to be rational.