Spectral synthesis with the complexity parameter

We show that spectral synthesis thresholds are governed by a quantitative spectral complexity parameter, the Fourier Ratio, in addition to the geometric size of the Fourier support. In the Euclidean setting, we prove that if a compactly supported measure has finite $\alpha$-dimensional packing measure and the associated Fourier ratio decays with asymptotic exponent $\kappa$, then the classical synthesis threshold improves from $\frac{2d}{\alpha }$ to $\frac{2(d-2\kappa )}{\alpha -2\kappa }$. We then establish an analogous result on compact Riemannian manifolds without boundary. In that setting the relevant object is a localized spectral Fourier ratio defined using Laplace–Beltrami spectral projectors. The resulting synthesis threshold is again determined by the decay exponent of this complexity parameter. These results place Euclidean and manifold spectral synthesis into a common framework in which geometric size and spectral complexity jointly govern uniqueness