A note about the deterministic property of characteristic functions

We study an extension property for characteristic functions f : R-n -> C of probability measures. More precisely, let f be the characteristic function of a probability density phi on R-n, and let U-sigma = {x is an element of R-n: min(k) |x(k)| > sigma}, sigma > 0, be a neighborhood of infinity. We say that f has the sigma-deterministic property if for any other characteristic function g such that f = g on U-sigma, it follows that f equivalent to g. A sufficient condition on f to has the sigma-deterministic property is given. We also discuss the question about how precise our sufficient condition is? These results show that the sigma-deterministic property of f depends on an arithmetic structure of the support of phi.