Spectral synthesis with the complexity parameter

TL;DR

This paper demonstrates that spectral synthesis thresholds are influenced by a spectral complexity parameter, the Fourier Ratio, alongside geometric support size, providing a unified framework for Euclidean and manifold settings.

Contribution

It introduces a spectral complexity parameter that refines synthesis thresholds, extending classical results to Riemannian manifolds using spectral projectors.

Findings

Spectral synthesis thresholds depend on the Fourier Ratio decay exponent.

Threshold improves from 2d/α to 2(d-2κ)/(α-2κ) with spectral complexity.

Results unify Euclidean and manifold spectral synthesis under a common framework.

Abstract

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