Midpoint-free subsets of the real numbers

A set of reals 𝑆 ⊂ R is midpoint-free if it has no subset {𝑎, 𝑏, 𝑐} ⊆ 𝑆 such that 𝑎<𝑏<𝑐 and 𝑎 + 𝑐 = 2𝑏. If 𝑆 ⊂ 𝑋 ⊆ R and 𝑆 is midpoint-free, it is a maximal midpoint-free subset of 𝑋 if there is no midpoint-free set 𝑇 such that 𝑆 ⊂ 𝑇 ⊆ 𝑋. In each of the cases 𝑋 = Z+, Z, Q+, Q, R+, R, we determine two maximal midpoint-free subsets of 𝑋 characterised by digit constraints on the base 3 representations of their members.