A note on powers of the characteristic function

Let $CH(R)$ denote the family of characteristic functions of probability measures (distributions) on the real line $R$. We study the following question: given an integer $n>1$, do there exist two different $f, g\in CH(R)$ such that $f^n\equiv g^n$? For positive even $n$, well-known examples answer this question in the affirmative. It turns out that the same is true also for any odd $n>1$. For $f\in CH(R)$ and integer $n>1$, set $C_n(f)=\{g\in CH(R): g^n\equiv f^n\}$. In this paper, we give an estimate for cardinality (or cardinal number) of $C_n(f)$. In addition, we describe such $f$ for which our estimate is sharp.