Metric entropy of Fourier ratio classes on ${\mathbb Z}_N$ (opens in new tab)

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 b...