Integral Diophantine approximation on varieties

We study the local behavior of integral points on log pairs near a fixed rational point in the boundary by means of an integral approximation constant. In light of Siegel's theorem about integral points on curves and McKinnon's conjecture on rational approximation constants, we conjecture that integral points that are close to the fixed point in archimedean topology should lie on certain rational curves with at most two points at infinity on weakly log Fano varieties. We verify this conjecture for a number of examples.