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Subset of reals (mod 1) with positive measure

Source: China TSTST 2017 Test 2 Day 2 Q6

March 13, 2017
algebracombinatoricsnumber theory

Problem Statement

Let MM be a subset of R\mathbb{R} such that the following conditions are satisfied:
a) For any x∈M,n∈Zx \in M, n \in \mathbb{Z}, one has that x+n∈Mx+n \in \mathbb{M}. b) For any x∈Mx \in M, one has that −x∈M-x \in M. c) Both MM and R\mathbb{R} \ MM contain an interval of length larger than 00.
For any real xx, let M(x)={n∈Z+∣nx∈M}M(x) = \{ n \in \mathbb{Z}^{+} | nx \in M \}. Show that if α,β\alpha,\beta are reals such that M(α)=M(β)M(\alpha) = M(\beta), then we must have one of α+β\alpha + \beta and α−β\alpha - \beta to be rational.
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