Abstract: I will report on recent results which deal with the manner in which diophantine arithmetic measures of distance to divisors relate to concepts of stability for polarized projective varieties. These results build on many previous insights including those of Boucksom-Chen, Evertse-Ferretti, K. Fujita, A. Levin, C. Li, McKinnon-Roth, Ru-Vojta, Ru-Wang and J. Silverman. Some emphasis will be placed on the case of K-(in)stability for Fano varieties. At the same time, I will present motivational examples which arise within the context of toric varieties.