Higher-order autocorrelation conjecture for super-resolution

Let $N$ observations of a signal be collected from the model, with each observation sampled at $L$ equally spaced locations; let $M$ denote the number of grid points and let $q\geq3$. Higher-order autocorrelation conjecture. In the low-SNR regime $\sigma\to\infty$, if $N/\sigma^{2q}\to\infty$, then one can identify up to $M=O(L^{q-1})$ grid points. Equivalently, $L=O(M^{1/(q-1)})$ samples per observation suffice for signal identification. In particular, when $N\to\infty$ and the noise level is fixed, however large, there is no theoretical limit on the achievable resolution. This conjecture extrapolates the third-order autocorrelation analysis to higher orders, which provide more polynomial equations as the number of observations increases.