We show that the problems of approximating the shortest and closest vector in a lattice to within a factor of √n lie in NP intersect CoNP. The result (almost) subsumes the three mutually-incomparable previous results regarding these lattice problems: Banaszczyk , Goldreich and Goldwasser , and Aharonov and Regev . Our technique is based on a simple fact regarding succinct approximation of functions using their Fourier series over the lattice. This technique might be useful elsewhere - we demonstrate this by giving a simple and efficient algorithm for one other lattice problem (CVPP) improving on a previous result of Regev . An interesting fact is that our result emerged from a "dequantization" of our previous quantum result in Aharonov and Regev . This route to proving purely classical results might be beneficial elsewhere.
- Fourier series