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

We show for decreasing, positive approximation functions ψ such that τ = lim q → ∞ -log ψ (q)/log q < 13 + √738 and such that q² ψ (q) → 0 that the set Exact(ψ ) of numbers approximable to the exact order ψ has positive Fourier dimension. This implies that the set Exact(ψ ) contains normal numbers.