A note on Bremner's conjecture and uniformity

In 1998, Bremner conjectured that elliptic curves over the rationals having long sequences of distinct rational points whose $x$-coordinates are in arithmetic progression, have large rank. This was proved some years ago in a strong form as a consequence of previous work by the authors, by a combination of Nevanlinna theory and the uniform Mordell--Lang theorem of Gao--Ge--K\"uhne. Thus, if the ranks of elliptic curves over the rationals are uniformly bounded, then so are the lengths of the aforementioned arithmetic progressions. In this note we give a much more direct proof of this last statement, using the height-uniform Mordell theorem of Dimitrov--Gao--Habegger. The method is flexible and we give a new application of these ideas to $x$-coordinates in finitely generated multiplicative groups and geometric progressions; connections to a possible semiabelian uniform Mordell--Lang are also discussed.