Some results on Lower Assouad and quantization dimensions

In this paper, we first show that the collection of all subsets of $\mathbb{R}$ having lower dimension $\gamma \in [0,1]$ is dense in $\mathrm{\Pi }(\mathbb{R})$, the space of compact subsets of $\mathbb{R}$. Furthermore, we show that the set of Borel probability measures with lower dimension $\beta \in [0,m]$ is dense in $\mathrm{\Omega }({\mathbb{R}}^m)$, the space of Borel probability measures on ${\mathbb{R}}^m$. We also prove that the quantization and the lower dimension of a measure $\vartheta$ coincide with those of the convolution of $\vartheta$ with a finite combination of Dirac measures. In the end, we compute the lower dimension of the invariant measure associated with the product IFS.