Positive sparse vectors in the kernel of a subsampled DFT matrix

Determine whether a dense random circulant graph's associated subsampled DFT matrix has a \(c n\)-sparse entrywise positive vector in its kernel for some sufficiently small constant \(c>0\).

Background

The proof of the paper's upper bound uses a norm inequality for vectors in the kernel of a subsampled discrete Fourier transform matrix. Existing lower bounds show that the kernel contains vectors with sparsity on the order of $n/\log n$, which prevents the authors from proving the sharp conjectured asymptotic by their present method. They nevertheless leave unresolved whether the additional entrywise-positivity constraint rules out vectors whose sparsity is a small constant fraction of $n$.