Elliptic curves and rational points in arithmetic progression

Abstract: Let $E/\mathbb{Q}$ be an elliptic curve. We consider finite sequences of rational points ${P_1,\ldots,P_N}$ whose $x$-coordinates form an arithmetic progression in $\mathbb{Q}$. Under the assumption of Lang's conjecture on lower bounds for canonical height functions, we prove that the length $N$ of such sequences satisfies the upper bound $\ll A^r$, where $A$ is an absolute constant and $r$ is the Mordell-Weil rank of $E/\mathbb{Q}$. Furthermore, assuming the uniform boundedness of ranks of elliptic curves over $\mathbb{Q}$, the length $N$ satisfies a uniform upper bound independent of $E$.