Spectral Factorization and Lattice Geometry

We obtain conditions for a trigonometric polynomial t of one variable to equal or be approximated by |p|^2 where p has frequencies in a Bohr set of integers obtained by projecting lattice points in the open planar region bounded by the lines y = alpha*x +- beta where |beta| leq 1/4 and alpha is either rational or irrational with Liouville-Roth constant larger than 2. We derive and use a generalization of the Fejer-Riesz spectral factorization lemma in one dimension, an approximate spectral factorization in two dimensions, the modular group action on the integer lattice, and Diophantine approximation.