Fourier Dimension Estimates for Sets of Exact Approximation Order: The Case of Small Approximation Exponents

Abstract 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 $\operatorname{Exact}(\psi )$ of numbers approximable to the exact order $\psi $ has positive Fourier dimension. This implies that the set $\operatorname{Exact}(\psi )$ contains normal numbers.