A conditional construction of restricted isometries
  We study the restricted isometry property of a matrix that is built from the discrete Fourier transform matrix by collecting rows indexed by quadratic residues. We find an $\epsilon>0$ such that, conditioned on a folklore conjecture in number theory, this matrix satisfies the restricted isometry property with sparsity parameter $K=\Omega(M^{1/2+\epsilon})$, where $M$ is the number of rows. 
Afonso S. Bandeira, Dustin G. Mixon, Joel Moreira
math.FA cs.IT math.IT math.NT