  Many results related to quantitative problems in the metric theory of
Diophantine approximation are asymptotic, such as the number of rational
solutions to certain inequalities grows with the same rate almost everywhere
modulo an asymptotic error term. The error term incorporates an implicit
constant that varies from one point to another. This means that applications of
these results does not give concrete bounds when applied to, say a finite sum,
or when applied to counting the number of solutions up to a finite point for a
given inequality. This paper addresses this problem and makes the tools and
their results effective, by making the implicit constant explicit outside of an
exceptional subset of Lebesgue measure at most $\delta>0$, an arbitrarily small
constant chosen in advance. We deduce from this the fully effective results for
Schmidt's Theorem, quantitative Koukoulopoulos-Maynard Theorem and quantitative
results on $M_{0}$-sets; we also provide effective results regarding statistics
of normal numbers and strong law of large numbers.
