Let Fq be the finite field of q elements having characteristic p, and denote by K_= Fq ( (1/t) ) the field of formal Laurent series in 1/t. We consider the equidistribution in T= K_/ Fqt of the values of polynomials f (u) K_ u as u varies over Fqt. Let K be a finite set of positive integers, and suppose that αᵣ K_ for r K \0\. We show that the polynomial ₑ ₊\₀\αᵣuʳ is equidistributed in T whenever αₖ is irrational for some k K satisfying p k, and also pᵛk K for any positive integer v. This conclusion resolves in full a conjecture made jointly by the third, fourth and fifth authors.
Champagne et al. (Thu,) studied this question.