Equidistribution criterion for polynomial sequences over function fields

Let $p$ be the characteristic of the finite field ${\mathbb F_q}$, let ${\mathbb K}_\infty$ be the associated completion, and let ${\mathbb T}$ be the relevant torus. Let

$$f(u)=\sum_{r \in {\mathcal K}\cup\{0\}} \alpha_r u^{r}$$

be a polynomial supported on ${\mathcal K}\subset {\mathbb Z}^+$, with coefficients $\alpha_r\in {\mathbb K}_\infty$. An element of ${\mathbb K}_\infty$ is called irrational when it is not rational in the relevant function-field sense. Equidistribution criterion. If $\alpha_k$ is irrational for some $k\in {\mathcal K}$ such that $p\nmid k$ and $p^v k\notin {\mathcal K}$ for every $v\in {\mathbb Z}^+$, then the sequence $(f(x))_{x\in {\mathbb F_q}[t]}$ is equidistributed in ${\mathbb T}$. The criterion is intended to isolate the obstructions arising from exponents divisible by the characteristic and from interference between Frobenius-related exponents; its resolution is not supplied in the source context.