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

The article focuses on establishing that sets of real numbers approximable to an exact order defined by a positive, decreasing approximation function \(\psi\), denoted \(\operatorname{Exact}(\psi)\), have positive Fourier dimension under certain conditions on \(\psi\). Specifically, if \(\psi\) satisfies \(q^{2}\psi(q) \to 0\) and the limit \(\tau = \lim_{q \to \infty} -\frac{\log \psi(q)}{\log q}\) exists with \(\tau < \frac{13 + \sqrt{73}}{8}\), then \(\operatorname{Exact}(\psi)\) supports a measure whose Fourier transform decays polynomially, implying positive Fourier dimension and the presence of normal numbers in \(\operatorname{Exact}(\psi)\). The authors construct such a measure via continued fraction expansions with mostly bounded partial quotients interspersed with sparse large partial quotients, and they analyze its geometric and oscillatory properties to prove the Fourier decay. This work extends previous results by enlarging the range of \(\tau\) for which positive Fourier dimension is known and provides detailed harmonic analysis and geometric measure theory arguments to support the main theorem.