If a d-dimensional proper GAP in Q contains a positive proportion ρ of the x-coordinates of rational points on an elliptic curve E of rank r, then the number of such points is at most A(E,d,ρ)^r.
Lang, Elliptic curves: Diophantine analysis
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Additive Rigidity for $x$-Coordinates of Rational Points on Elliptic Curves
If a d-dimensional proper GAP in Q contains a positive proportion ρ of the x-coordinates of rational points on an elliptic curve E of rank r, then the number of such points is at most A(E,d,ρ)^r.