According to Mattila, *Geometry of sets and measures in Euclidean spaces*, p. 168, the **Fourier dimension** $\text{dim}_F(A)$ of $A\subseteq \mathbb R^n$ is the unique number in $[0,n]$ such that for any $0<s<\text{dim}_F A$ there exists a non-zero Radon measure $\mu$ with spt $\mu\subset A$ and $|\hat\mu(x)|\le |x|^{-s/2}$ for $x\in\mathbb R^n$, and that for $\text{dim}_F(A)<s\le n$ no such measure exists.

Does this condition on $\mu$ for $\text{dim}_F(A)>0$ imply a nontrivial lower bound on the $\mu$-measure of a ball of radius $r$, $\mu(B(x,r))$ in terms of $r$, for $\mu$-almost all $x$? For instance, a lower bound like $r^s$ for a constant $s$?

(The reason for asking being that positive Hausdorff dimension is enough to get such an *upper* bound and I'm wondering if one can get *lower* bounds by strengthening the assumption "positive Hausdorff dimension" to "positive Fourier dimension" or something similar.)