  For any family of elliptic curves over the rational numbers with fixed
$j$-invariant, we prove that the existence of a long sequence of rational
points whose $x$-coordinates form a non-trivial arithmetic progression implies
that the Mordell-Weil rank is large, and similarly for $y$-coordinates. We give
applications related to uniform boundedness of ranks, conjectures by Bremner
and Mohanty, and arithmetic statistics on elliptic curves. Our approach
involves Nevanlinna theory as well as R\'emond's quantitative extension of
results of Faltings.
