If $X$ is a variety of general type defined over a number field $k$, then the Bombieri-Lang conjecture predicts that the $k$-rational points of $X$ are not Zariski dense. The conjecture is a prediction that a global condition on the canonical bundle (that it is “generically positive”) implies a global condition about rational points. By the local-global philosophy in geometry we should look for local influence of positivity on the accumulation of rational points. To do that we need measures of both these local phenomena. \
Let $L$ be an ample line bundle on $X$, and $x\in X(\overline{k})$. The central theme of the talk is the interrelations between $\alpha_x(L)$, an invariant measuring the accumulation of rational points around $x$ as gauged by $L$, and the Seshadri constant $\epsilon_{x}(L)$, measuring the local positivity of $L$ near $x$. In particular, the classic approximation theorem of K. F. Roth on $\mathbf{P}^1$ generalizes as an inequality between $\alpha_{x}$ and $\epsilon_{x}$ valid for all projective varieties. \
This is joint work with David McKinnon.