A Weyl-type equidistribution theorem in finite characteristic
type: Item
instance of: scholarly article
title: A Weyl-type equidistribution theorem in finite characteristic
zbMATH Open document ID: 1336.37003
author: V. Bergelson
author: A. Leibman
published in: Advances in Mathematics
publication date: 2016-01-15T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 37A05
Mathematics Subject Classification ID: 37A30
Mathematics Subject Classification ID: 28D10
zbMATH DE Number: 6530928
zbMATH Keywords: Weyl equidistribution
zbMATH Keywords: positive characteristic
zbMATH Keywords: Følner sequence
reviewed by: Thomas B. Ward
MaRDI profile type: Publication
full work available at URL: j.aim.2015.11.027
OpenAlex ID: W2190870480
cites work: Ergodic theory and diophantine problems
cites work: Polynomial Szemerédi theorems for countable modules over integral domains and finite fields
cites work: A Roth theorem for amenable groups.
cites work: Diophantine Approximation in Fields of Characteristic p
cites work: Q5574757
cites work: Q5597651
cites work: On uniform distribution of sequences in \(GF[q,x]\) and \(GF\{q,x\}\)
cites work: Sequences, discrepancies and applications
cites work: Q3909268
cites work: Van der Corput's difference theorem
cites work: Q3684180
cites work: On difference sets of sequences of integers. I
DOI: 10.1016/J.AIM.2015.11.027
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description: scientific article; zbMATH DE number 6530928