# Dimension and measures on sub-self-affine sets

## TL;DR

This paper investigates the dimensions of sub-self-affine sets, showing they typically match and equal the zero of a topological pressure, and explores properties of measures supported on these sets.

## Contribution

It provides a partial positive answer to Falconer's question by establishing the dimension equality and analyzes the topological pressure and measure properties.

## Key findings

- Hausdorff and Minkowski dimensions coincide in typical sub-self-affine sets.
- Dimensions equal the zero of an appropriate topological pressure.
- Properties of natural measures on sub-self-affine sets are characterized.

## Abstract

We show that in a typical sub-self-affine set, the Hausdorff and the Minkowski dimensions coincide and equal the zero of an appropriate topological pressure. This gives a partial positive answer to the question of Falconer. We also study the properties of the topological pressure and the existence and the uniqueness of natural measures supported on a sub-self-affine set.