turkmath.org

The well-known Wiener theorem states that if the function $f$ is bounded away from zero on $[0, 2π]$ and has an absolutely convergent Fourier series expansion, then the function $\dfrac{1}{f}$ has the same property. A local version of this theorem is proved without the condition of boundedness away from zero. It also extends to Dirichlet series. Applications: 1) in the theory of quasicrystals (a new sufficient condition for the representability of a discrete measure support in Euclidean space as the union of a finite number of lattices) 2) a new sufficient condition for a discrete set in Euclidean space to be a set of coherent frequencies