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.)