Diophantine approximation with perfect squares and the solvability of an inhomogeneous wave equation

Diophantine criteria occur naturally in the theory of partial differential equations through the notorious problem of small denominators. An extensive treatment of such problems in the theory of PDEs can be found, e.g., in [6]. In this paper, we are interested in a Diophantine problem related to an inhomogeneous wave equation in n spatial and one temporal dimension with periodic boundary conditions. In brief, to ensure the convergence of a formal solution to the equation certain conditions on the periods should be satisfied. These conditions normally leave a small set of exceptional periods for which the convergence of the series is problematic, though the solution might exist. It is therefore of interest to measure the ‘size’ of the exceptional set of periods. Regarding the wave equation we will discuss the problem in more details and derive the associated Diophantine problem in §2.