Elliptic curves and rational points in arithmetic progression

Let E/Q be an elliptic curve. We consider finite sequences of rational points ₁,…,PN\ whose x-coordinates form an arithmetic progression in 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 Aʳ, where A is an absolute constant and r is the Mordell-Weil rank of E/Q. Furthermore, assuming the uniform boundedness of ranks of elliptic curves over Q, the length N satisfies a uniform upper bound independent of E.