  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, if the (integer) oversampling ratio $\lambda$ is 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, then for any number $K\geq 2$ of
quantization levels available for the real and imaginary parts of the
measurements, our quantization method guarantees reconstruction accuracy of
order $O(\lambda K^{- \lambda/2})$, up to constants which are independent of
$K$ and $\lambda$. In contrast, memoryless scalar quantization offers a
guarantee of order $O(K^{-1})$ only.
