Study on Uniform Distribution in Positive Characteristic

Abstract

In classical Diophantine approximation, uniform distribution is an important topic. The distribution of real numbers and the estimation of exponential sums using Weyl's criteria are inextricably linked.Carlitz provided an elementary definition of uniform distribution in positive characteristics (see [1]), but we will find a geometrical description.In this paper, we use the Haar measure to present a precise analogue to Weyl's criteria in the case of positive characteristics.  As an application, we demonstrate that the uniformly distributed modulo 1 for linear forms and polynomial functions.We show that in the Laurent series field, the set {m\(\theta\)}  is uniformly distributed modulo 1, where m extends over all polynomials and \(\theta\)  is a fixed irrational function.