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

2026-06-23

AI Summary

This study investigates the metric entropy and uniform sampling properties of signal classes on ${\mathbb Z}_N$ with a prescribed Fourier ratio $r$. By interpreting the squared Fourier ratio $FR(f)^2$ as an effective dimensionality parameter, the authors combine metric entropy estimates, covering number analysis, and $\ell^2$-separation arguments for phase orbits to establish matching upper and lower bounds on metric entropy at small scales, leading to uniform error bounds for empirical approximation. The primary contribution lies in systematically revealing, for the first time, the decisive role of the Fourier ratio in governing the geometric complexity, approximability, and phase-orbit separability of such signal classes. Furthermore, the work constructs an exponentially large family of positively separated signals, thereby providing theoretical support for uniform sampling frameworks.

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.

Problem

• metric entropy
• Fourier ratio
• uniform sampling
• signal classes
• spectral distribution

Innovation

• metric entropy
• Fourier ratio
• uniform sampling
• phase-orbit packing
• effective dimension