${\mathbb{A}}^1$-connected varieties over non-closed fields

In this paper, we proved two results regarding the arithmetics of separably ${\mathbb{A}}^1$-connected varieties of rank one. First we proved over a large field, there is an ${\mathbb{A}}^1$-curve through any rational point of the boundary, if the boundary divisor is smooth and separably rationally connected. Secondly, we generalize a theorem of Hassett-Tschinkel for the Zariski density of integral points over function fields of curves.