Generalizing a result of Bombieri, Masser, and Zannier we show that on a curve in the algebraic torus which is not contained in any proper coset only finitely many points are close to an algebraic subgroup of codimension at least $2$. The notion of close is defined using the Weil height. We also deduce some cardinality bounds and further finiteness statements.

Let $K$ be a global field of characteristic not 2. Let $\mathbf{Z}=\mathbf{H}\setminus \mathbf{G}$ be a symmetric variety defined over $K$ and $S$ a finite set of places of $K$. We obtain counting and equidistribution results for the S-integral points of $\mathbf{Z}$. Our results are effective when $K$ is a number field.

C. J. Smyth was among the first to study the spectrum of the Weil height in the field of all totally real numbers, establishing both lower and upper bounds for the limit infimum of the height of all totally real integers, and determining isolated values of the height. Later, Bombieri and Zannier established similar results for totally p-adic numbers and, inspired by work of Ullmo and Zhang, termed this the Bogomolov property. In this paper, we use results on equidistribution of points of low height...

We study the behavior of canonical height functions $h{̂}_f$, associated to rational maps f, on totally p-adic fields. In particular, we prove that there is a gap between zero and the next smallest value of $h{̂}_f$ on the maximal totally p-adic field if the map f has at least one periodic point not contained in this field. As an application we prove that there is no infinite subset X in the compositum of all number fields of degree at most d such that f(X) = X for some non-linear polynomial f. This answers a...

Bombieri and Zannier established lower and upper bounds for the limit infimum of the Weil height in fields of totally $p$-adic numbers and generalizations thereof. In this paper, we use potential theoretic techniques to generalize the upper bounds from their paper and, under the assumption of integrality, to improve slightly upon their bounds.