Title: Exact Hausdorff and packing measures for random self-similar code-trees with necks,
Abstract: Random code-trees with necks were introduced recently to generalise the
notion of $V$-variable and random homogeneous sets. While it is known that the
Hausdorff and packing dimensions coincide irrespective of overlaps, their exact
Hausdorff and packing measure has so far been largely ignored. In this article
we consider the general question of an appropriate gauge function for positive
and finite Hausdorff and packing measure. We first survey the current state of
knowledge and establish some bounds on these gauge functions. We then show that
self-similar code-trees do not admit a gauge functions that simultaneously give
positive and finite Hausdorff measure almost surely. This surprising result is
in stark contrast to the random recursive model and sheds some light on the
question of whether $V$-variable sets interpolate between random homogeneous
and random recursive sets. We conclude by discussing implications of our
results.