Diophantine approximation with perfect squares and the solvability of an   inhomogeneous wave equation
  The Hausdorff dimension of an exceptional set of periods for which convergence of a formal solution to an inhomogeneous wave equation in n spatial and one temporal dimension is problematic, is determined along with conditions which the periods must satisfy to ensure the solvability of the inhomogeneous wave equation by a smooth periodic function. To derive this information, a complete metric theory for a related fully nonlinear Diophantine approximation problem involving perfect squares is established. 
V. Beresnevich, M. Dodson, S. Kristensen, J. Levesley