  Until recently, it was an important open problem in Fractal Geometry to
determine whether there exists an iterated function system acting on
$\mathbb{R}$ with no exact overlaps for which cylinders are super-exponentially
close at all small scales. Iterated function systems satisfying these
properties were shown to exist by the author and by B\'{a}r\'{a}ny and
K\"{a}enm\"{a}ki. In this paper we prove a general theorem on the existence of
such iterated function systems within a parameterised family. This theorem
shows that if a parameterised family contains two independent subfamilies, and
the set of parameters that cause exact overlaps satisfies some weak topological
assumptions, then the original family will contain an iterated function system
satisfying the desired properties. We include several explicit examples of
parameterised families to which this theorem can be applied.
