Equidistribution Modulo 1

Abstract

The generalisation of questions of the classic arithmetic has long been of interest. One line of questioning, introduced by Car in 1995, inspired by the equidistribution of the sequence ${\left(n^{\alpha }\right)}_{n\in N}$ where $0<\alpha <1$ , is the study of the sequence $\left(K^{\left(1/l\right)}\right)$ , where $K$ is a polynomial having an l-th root in the field of formal power series. In this paper, we consider the sequence $\left(L^{\prime \left(1/l\right)}\right)$ , where $L\prime$ is a polynomial having an l-th root in the field of formal power series and satisfying $L\prime \equiv B\mathrm{mod}C$ . Our main result is to prove the uniform distribution in the Laurent series case. Particularly, we prove the case for irreducible polynomials.