  In previous work, the authors established a generalized version of Schmidt's
subspace theorem for closed subschemes in general position in terms of Seshadri
constants. We extend our theorem to weighted sums involving closed subschemes
in subgeneral position, providing a joint generalization of Schmidt's theorem
with seminal inequalities of Nochka. A key aspect of the proof is the use of a
lower bound for Seshadri constants of intersections from algebraic geometry, as
well as a generalized Chebyshev inequality. As an application, we extend
inequalities of Nochka and Ru-Wong from hyperplanes in $m$-subgeneral position
to hypersurfaces in $m$-subgeneral position in projective space, proving a
sharp result in dimensions $2$ and $3$, and coming within a factor of $3/2$ of
a sharp inequality in all dimensions. We state analogous results in Nevanlinna
theory generalizing the Second Main Theorem and Nochka's theorem (Cartan's
conjecture).
