Exact Hausdorff and packing measures for random self-similar code-trees with necks

TL;DR

This paper investigates the exact Hausdorff and packing measures of random self-similar code-trees with necks, revealing that they do not admit gauge functions that produce positive and finite measures, contrasting with other models.

Contribution

It provides the first detailed analysis of gauge functions for these measures and shows a surprising non-existence result for self-similar code-trees.

Findings

Hausdorff and packing dimensions coincide regardless of overlaps

Self-similar code-trees lack gauge functions for positive finite measures

Contrasts with the behavior of random recursive models

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.