# Lower Assouad Dimension of Measures and Regularity

## TL;DR

This paper investigates the lower Assouad dimensions of measures, exploring their relationship with other dimensions and regularity, and establishing results for self-similar and self-affine measures.

## Contribution

It introduces the concept of lower Assouad dimension for measures, analyzes its properties, and establishes conditions under which it equals other dimensions, especially for self-similar measures.

## Key findings

- Lower Assouad dimension is dominated by the infimum of local dimensions.
- Self-affine measures are uniformly perfect with positive lower Assouad dimension.
- The Assouad spectrum converges to the quasi-Assouad dimension.

## Abstract

In analogy with the lower Assouad dimensions of a set, we study the lower Assouad dimensions of a measure. As with the upper Assouad dimensions, the lower Assouad dimensions of a measure provide information about the extreme local behaviour of the measure. We study the connection with other dimensions and with regularity properties. In particular, the quasi-lower Assouad dimension is dominated by the infimum of the measure's lower local dimensions. Although strict inequality is possible in general, equality holds for the class of self-similar measures of finite type. This class includes all self-similar, equicontractive measures satisfying the open set condition, as well as certain `overlapping' self-similar measures, such as Bernoulli convolutions with contraction factors that are inverses of Pisot numbers. We give lower bounds for the lower Assouad dimension for measures arising from a Moran construction, prove that self-affine measures are uniformly perfect and have positive lower Assouad dimension, prove that the Assouad spectrum of a measure converges to its quasi-Assouad dimension and show that coincidence of the upper and lower Assouad dimension of a measure does not imply that the measure is $s$-regular.