# Measures with specified support and arbitrary Assouad dimensions

## TL;DR

This paper demonstrates that for any compact set in the real line with positive upper Assouad dimension, one can construct measures supported on it with prescribed upper or lower Assouad dimensions, extending the understanding of measure-dimension relationships.

## Contribution

It establishes the existence of measures with arbitrary prescribed upper or lower Assouad dimensions supported on sets with positive upper Assouad dimension.

## Key findings

- Existence of measures with any upper Assouad dimension greater than the set’s dimension.
- Existence of measures with any lower Assouad dimension less than the set’s lower dimension.
- Extension of measure construction techniques to arbitrary Assouad dimensions.

## Abstract

We show that if the upper Assouad dimension of the compact set $E\subseteq \mathbb{R}$ is positive, then given any $D>\dim_{A}E$ there is a measure with support $E$ and upper Assouad (or regularity) dimension $D$. Similarly, given any $0\leq d<\dim_{L}E,$ there is a measure on $E$ with lower Assouad dimension $d$.