Fourier Dimension Estimates for Sets of Exact Approximation Order: The Badly-Approximable Case

We show for decreasing, positive approximation functions $\psi$ such that $\tau ={\lim }_{q\to \mathrm{\infty }}\frac{\log \psi (q)}{\log q}<\frac{13+\sqrt{73}}{8}$ 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.