“…It is a fundamental theorem of Veech [119] that the structure relation for the maximal equicontinuous factor map g is regional proximality: tilings T , T ∈ Ω Φ are regionally proximal, T ∼ rp T , if and only if, for each > 0 there are S, S ∈ Ω Φ and v ∈ R n so that: (i) d(T , S) < , (ii) d(T , S ) < , and (iii) d(S − v, S − v) < . (For a general discussion of regional proximality in tiling spaces see [28].) In the context of Theorem 7.7 one can show that T ∼ gs T =⇒ T ∼ rp T .…”