Binary Discrete Fourier Transform and Its Inversion

A binary vector of length $N$ has elements that are either 0 or 1. We investigate the question whether and how a known can be reconstructed from limited set its discrete Fourier transform (DFT) coefficients. priori information is provides powerful constraint. prove uniquely defined by two complex DFT coefficients (zeroth, which gives popcount, first) if prime. If prime factors, additional must included in data to guarantee uniqueness, we find number required theoretically. One may need know even more stability inversion. However, our results indicate stable inversion obtained when about $1/3$ total. This entails effect super-resolution (the resolution limit improved factor $\sim 3$).