  We generalize Waldschmidt's bound for Leopoldt's defect and prove a similar
bound for Gross's defect for an arbitrary extension of number fields. As an
application, we prove new cases of Gross's finiteness conjecture (also known as
the Gross-Kuz'min conjecture) beyond the classical abelian case, and we show
that Gross's $p$-adic regulator has at least half of the conjectured rank. We
also describe and compute non-cyclotomic analogues of Gross's defect.
