Direct proof via height-uniform Mordell theorem shows uniform rank bounds for elliptic curves over Q imply uniform bounds on lengths of arithmetic progressions in x-coordinates of rational points, with extensions to multiplicative groups and geometric progressions.
Bombieri,The Mordell conjecture revisited
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A note on Bremner's conjecture and uniformity
Direct proof via height-uniform Mordell theorem shows uniform rank bounds for elliptic curves over Q imply uniform bounds on lengths of arithmetic progressions in x-coordinates of rational points, with extensions to multiplicative groups and geometric progressions.