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...