  We explain how complexity of rational points on projective varieties can be
interpreted via the theories of Chow forms and Okounkov bodies. Precisely, we
study discrete measures on filtered linear series and build on work of Boucksom
and Chen, Boucksom-et-al and Ferretti. For example, we show how asymptotic
volume and Seshadri constants are related to the theories of Chow weights and
Okounkov bodies for very ample linear series. We also consider arithmetic
applications within the context of Diophantine approximation. In this
direction, we establish Roth-type theorems which can be expressed in terms of
normalized Chow weights and measures of local positivity.
