# Separation conditions on controlled Moran constructions

## TL;DR

This paper extends the equivalence between the open set condition and positive Hausdorff measure from self-similar sets to a broader class of Moran constructions, analyzing various types of these constructions.

## Contribution

It establishes a new equivalence for Moran constructions, generalizing known results from self-similar sets and exploring different Moran structures.

## Key findings

- Proves the equivalence for a class of Moran constructions.
- Analyzes various types of Moran constructions.
- Extends known results from self-similar sets.

## Abstract

It is well known that the open set condition and the positivity of the $t$-dimensional Hausdorff measure are equivalent on self-similar sets, where $t$ is the zero of the topological pressure. We prove an analogous result for a class of Moran constructions and we study different kinds of Moran constructions with this respect.