Metric entropy of Fourier ratio classes on ${\mathbb{Z}}_N$

Abstract

We study metric entropy and uniform sampling for classes of signals on ${\mathbb{Z}}_N$ with prescribed Fourier ratio. The Fourier ratio measures how spread out the Fourier transform of a signal is, interpolating between sparse spectral support and nearly uniform spectral distribution. Our main result gives upper and lower bounds for the metric entropy of a Fourier-ratio layer of size $r.$ At any sufficiently small fixed covering scale, these bounds match in their dependence on $r$ and $N$ and show that $FR(f{)}^2$ acts as an effective dimension parameter governing the size of the class. We use the entropy estimate to obtain uniform bounds for empirical approximation over Fourier-ratio classes. We also establish a phase-orbit packing result. If a single signal has a flat spectral block of size $k,$ then phase perturbations of that signal generate an exponentially large family with the same Fourier ratio and positive ${\ell }^2$ separation. Together, these results show that the Fourier ratio governs not only approximation properties of individual signals, but also the geometric size and uniform sampling behavior of entire signal classes.