Additive Rigidity for $x$-Coordinates of Rational Points on Elliptic Curves

We study the interaction between the group law on an elliptic curve and the additive structure of $x$-coordinates of rational points on an elliptic curve. Let $E\mathrm{/}\mathbb{Q}$ be an elliptic curve of Mordell-Weil rank $r\ge 1$, $d\ge 1$ be an integer, and $0<\rho \le 1$. We show that if a $d$-dimensional proper generalized arithmetic progression in $\mathbb{Q}$ contains the $x$-coordinates of rational points on $E\mathrm{/}\text{bbq}$ with positive proportion $\rho$, then the number of such points is bounded by $A(E,d,\rho {)}^r$. The proof combines extraction lemmas, gap principles, and the bounds for spherical codes. As an application, we obtain restrictions on sets of rational points whose $x$-coordinates have small sumsets or large additive energy.