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We show that if a $d$-dimensional proper generalized arithmetic progression in $\\mathbb{Q}$ contains the $x$-coordinates of rational points on $E/\\bbq$ with positive proportion $\\rho$, then the number of such points is bounded by $A(E,d,\\rho)^r$. 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Let $E/\\mathbb{Q}$ be an elliptic curve of Mordell-Weil rank $r \\geq 1$, $d \\geq 1$ be an integer, and $0<\\rho \\leq 1$. We show that if a $d$-dimensional proper generalized arithmetic progression in $\\mathbb{Q}$ contains the $x$-coordinates of rational points on $E/\\bbq$ with positive proportion $\\rho$, then the number of such points is bounded by $A(E,d,\\rho)^r$. 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