Let $K$ be a number field and let $h$ be a sufficiently nice height on the projective line over $\bar{K}$. A celebrated theorem of Favre and Rivera-Letelier states that the Galois orbits of an $h$-small sequence of distinct points will equidistribute at each place of $K$. In particular, it is necessary that such a sequence be of bounded height.

Could this also be sufficient, at least along a subsequence? That is, if $\alpha_n$ is a sequence of bounded height, then can one find a subsequence $\alpha_{n_k}$ and a nice height $h$ so that $h(\alpha_{n_k}) \to 0$? We’ll discuss this question and some potential applications. Joint work-in-progress with Paul Fili.