Positive sparse kernel vectors for subsampled DFT matrices

Determine whether a sufficiently small constant c>0 exists such that no cn-sparse entrywise positive vector belongs to the kernel of the random subsampled discrete Fourier transform matrix associated with a dense random circulant graph.

Background

The proof strategy for the sharp √n conjecture would require stronger control of vectors in the kernel of the subsampled DFT matrix. Existing lower bounds for restricted isometry permit sparse kernel vectors, but the paper notes that these vectors need not be entrywise positive. Resolving whether positive sparse kernel vectors exist could clarify whether the restricted-isometry approach can prove the conjectured sharp asymptotic.