  In this paper, we consider the extensively studied problem of computing a
$k$-sparse approximation to the $d$-dimensional Fourier transform of a length
$n$ signal. Our algorithm uses $O(k \log k \log n)$ samples, is dimension-free,
operates for any universe size, and achieves the strongest $\ell_\infty/\ell_2$
guarantee, while running in a time comparable to the Fast Fourier Transform. In
contrast to previous algorithms which proceed either via the Restricted
Isometry Property or via filter functions, our approach offers a fresh
perspective to the sparse Fourier Transform problem.
