  We show that the method of distributed noise-shaping beta-quantization offers
superior performance for the problem of spectral super-resolution with
quantization whenever there is redundancy in the number of measurements. More
precisely, we define the oversampling ratio $\lambda$ as the largest integer
such that $\lfloor M/\lambda\rfloor - 1\geq 4/\Delta$, where $M$ denotes the
number of Fourier measurements and $\Delta$ is the minimum separation distance
associated with the atomic measure to be resolved. We prove that for any number
$K\geq 2$ of quantization levels available for the real and imaginary parts of
the measurements, our quantization method combined with either TV-min/BLASSO or
ESPRIT guarantees reconstruction accuracy of order $O(M^{1/4}\lambda^{5/4} K^{-
\lambda/2})$ and $O(M^{3/2} \lambda^{1/2} K^{- \lambda})$ respectively, where
the implicit constants are independent of $M$, $K$ and $\lambda$. In contrast,
naive rounding or memoryless scalar quantization for the same alphabet offers a
guarantee of order $O(M^{-1}K^{-1})$ only, regardless of the reconstruction
algorithm.
