Deterministic sparse fourier approximation via approximating arithmetic progressions

We present a deterministic algorithm for finding the significant Fourier frequencies of a given signal f ∈ C ^N and their approximate Fourier coefficients in running time and sample complexity polynomial in N, L₁(f̂)/||f̂₂, and 1/τ , where the significant frequencies are those occupying at least a τ-fraction of the energy of the signal, and L₁(f̂) denotes the L₁-norm of the Fourier transform of f. Furthermore, the algorithm is robust to additive random noise. This strictly extends the class of compressible/Fourier sparse signals efficiently handled by previous deterministic algorithms for signals in C^N. As a central tool, we prove there is a deterministic algorithm that takes as input N , ε and an arithmetic progression P in Z_N, runs in time polynomial in ln N and 1/ε, and returns a set A_P that ε-approximates P in Z_N in the sense that equation presented}. In other words, we show there is an explicit construction of sets A_P of size polynomial in ln N and 1/ε that ε-approximate given arithmetic progressions P in Z_N. This extends results on small-bias sets, which are sets approximating the entire domain, to sets approximating a given arithmetic progression; this result may be of independent interest.