  We show for decreasing, positive approximation functions $\psi$ such that
$\tau = \lim_{q \to \infty} \frac{\log \psi(q)}{\log q} < \frac{13 +
\sqrt{73}}{8}$ and such that $q^2 \psi(q) \to 0$ that the set
$\text{Exact}(\psi)$ of numbers approximable to the exact order $\psi$ has
positive Fourier dimension. This implies that the set $\text{Exact}(\psi)$
contains normal numbers.
