  We consider the problem of robustly recovering a $k$-sparse coefficient
vector from the Fourier series that it generates, restricted to the interval
$[- \Omega, \Omega]$. The difficulty of this problem is linked to the
superresolution factor SRF, equal to the ratio of the Rayleigh length (inverse
of $\Omega$) by the spacing of the grid supporting the sparse vector. In the
presence of additive deterministic noise of norm $\sigma$, we show upper and
lower bounds on the minimax error rate that both scale like $(SRF)^{2k-1}
\sigma$, providing a partial answer to a question posed by Donoho in 1992. The
scaling arises from comparing the noise level to a restricted isometry constant
at sparsity $2k$, or equivalently from comparing $2k$ to the so-called
$\sigma$-spark of the Fourier system. The proof involves new bounds on the
singular values of restricted Fourier matrices, obtained in part from old
techniques in complex analysis.
