Remarks on R\'emond's generalized Lehmer problems

Abstract

We draw connections between the various conjectures which are included in G. R\'emond's generalized Lehmer problems. Specifically, we show that the degree one form of his conjecture for the multiplicative group is, in a sense, almost as strong as the strong form. Our insights into the conjectures apply a basic result on what may be called Diophantine approximation in the metric induced by the canonical height. We also present a previously unpublished partial result of R\'emond on his conjectures.