Let $f(T) = \sum a_n T^n \in \mathbf{F}_p [[ T ]]$ be a power series. We'll say that $f(T)$ is equidistributed if for every $a \in \mathbf{F}_p$, we have
$$\lim_{X \to \infty} \dfrac{1}{X} \cdot \# \{n<X : a_n =a \} = \dfrac{1}{p}.$$
Suppose that $f(T), g(T) \in \mathbf{F}_p [[ T ]]$ are power series which are equidistributed in the above sense. It is not true in general that the product $f(T) \cdot g(T)$ is equidistributed. (E.g: see this question for counterexamples.) But are there sufficient conditions we can place on $f$ and $g$ that ensure that the product $f(T) \cdot g(T)$ is equidistributed?
