  We continue with our study of the arithmetic geometry of toric varieties. In
this text, we study the positivity properties of metrized R-divisors in the
toric setting. For a toric metrized R-divisor, we give formulae for its
arithmetic volume and its chi-arithmetic volume, and we characterize when it is
arithmetically ample, nef, big or pseudo-effective, in terms of combinatorial
data. As an application, we prove a Dirichlet's unit theorem on toric
varieties, we give a characterization for the existence of a Zariski
decomposition of a toric metrized R-divisor, and we prove a toric arithmetic
Fujita approximation theorem.
