In the Sparse Fourier Transform problem, we are given oracle access to a vector x, and the goal is to compute the k largest in magnitude Fourier coefficients of x. Typically the goal is to minimize the number of samples (oracle accesses) and the running time of the algorithm. Our focus in this talk will be the sample complexity (#oracle acceses) in the d-dimensional case, for which previous work either suffered from an exponential dependence on d, or recovered the k largest Fourier coefficients only in an average sense. What we will show is that obtaining the best of both worlds is possible; the crux of our argument is a novel technique for re-using samples in (a specific form of) iterative procedures.