A Weyl-type equidistribution theorem in finite characteristic
An analogue of Weyl's polynomial equidistribution theorem on the torus is formulated and proved in positive characteristic. To a finite field \(F\) is associated a ring \(Z=F[t]\) with field of fractions \(Q=F(t)\) whose completion with respect to the so-called infinite valuation is \(R=F((t^{-1}))\), giving rise to an analogue of the \(d\)-torus in the form \(T^d=(R/Z)^d\). The main result is that for any element \(\alpha=(\alpha_1,\dots,\alpha_d)\in T^d\) the \(Z\)-sequence \((\alpha n\mid n\in Z)\) is well distributed (a stronger form of equidistribution expressed in terms of sequence along any Følner sequence in \(Z\)) in a subgroup of the form \(S(\alpha)+K\alpha\), where \(S(\alpha)\) is a certain kind of sub-torus of \(T^d\), \(K\) is a finite subset of \(Z\), and \(S(\alpha)=T^c\) if and only if the components \(\alpha_1,\dots,\alpha_d\) are \(Z\)-linearly independent.
scientific article; zbMATH DE number 6530928
scientific article