The Fourier Ratio and complexity of signals

Abstract: We study the Fourier ratio of a signal $f:\mathbb Z_N\to\mathbb C$, [ \mathrm{FR}(f)\ :=\ \sqrt{N}\,\frac{|\widehat f|{L^1(μ)}}{|\widehat f|{L^2(μ)}} \ =\ \frac{|\widehat f|_1}{|\widehat f|_2}, ] as a simple scalar parameter governing Fourier-side complexity, structure, and learnability. Using the Bourgain--Talagrand theory of random subsets of orthonormal systems, we show that signals concentrated on generic sparse sets necessarily have large Fourier ratio, while small $\mathrm{FR}(f)$ forces $f$ to be well-approximated in both $L^2$ and $L^\infty$ by low-degree trigonometric polynomials. Quantitatively, the class ${f:\mathrm{FR}(f)\le r}$ admits degree $O(r^2)$ $L^2$-approximants, which we use to prove that small Fourier ratio implies small algorithmic rate--distortion, a stable refinement of Kolmogorov complexity.