  In this paper, we relate the framework of mod-$\phi$ convergence to the
construction of approximation schemes for lattice-distributed random variables.
The point of view taken here is that of Fourier analysis in the Wiener algebra,
allowing the computation of asymptotic equivalents in the local, Kolmogorov and
total variation distances. By using signed measures instead of probability
measures, we are able to construct better approximations of discrete lattice
distributions than the standard Poisson approximation. This theory applies to
various examples arising from combinatorics and number theory: number of cycles
in (possibly coloured) permutations, number of prime divisors (possibly within
different residue classes) of a random integer, number of irreducible factors
of a random polynomial, etc. One advantage of the approach developed in this
paper is that it allows us to deal with approximations in higher dimensions as
well. In this setting, we can explicitly see the influence of the correlations
between the components of the random vectors in our asymptotic formulas.
