A note about the deterministic property of characteristic functions

We study an extension property for characteristic functions f : Rn → C of probability measures. More precisely, let f be the characteristic function of a probability density φ on Rn, and let Uσ = {x ∈ Rn: mink|xk| > σ}, σ > 0, be a neighborhood of infinity. We say that f has the σ-deterministic property if for any other characteristic function g such that f = g on Uσ, it follows that f ≡ g. A sufficient condition on f to has the σ-deterministic property is given. We also discuss the question about how precise our sufficient condition is? These results show that the σ-deterministic property of f depends on an arithmetic structure of the support of φ.