  For $m\ge 2$, consider $K$ the $m$-fold Cartesian product of the limit set of
an IFS of two affine maps with rational coefficients. If the contraction rates
of the IFS are reciprocals of integers, and $K$ does not degenerate to
singleton, we construct vectors in $K$ that lie within the ``folklore set'' as
defined by Beresnevich et al., meaning they are Dirichlet improvable but not
singular or badly approximable (in fact our examples are Liouville vectors). We
further address the topic of lower bounds for the Hausdorff and packing
dimension of these folklore sets within $K$, however we do not compute bounds
explicitly. Our class of fractals extends (Cartesian products of) classical
missing digit fractals, for which analogous results had recently been obtained.
