Bounded height in pencils of subgroups of finite rank

In a recent paper we proved some new bounded height results for equations involving varying integer exponents. Here we make a start on the problem of generalizing to rational exponents, which corresponds to the step from groups that are finitely generated to groups of finite rank. We discover two unexpected obstacles. The first is that bounded height may genuinely fail in the neighbourhood of certain exponents. The second concerns vanishing subsums, which seem to be much harder to deal with than in classical situations like S-unit equations. But for certain simple and natural equations we are able to clarify the first obstacle and eliminate the second. The proofs are partly based on our earlier work but there are also new considerations about successive minima over function fields.