Equidistribution Modulo 1

Wiem Gadri,Shaofang Hong

doi:10.1155/2021/1357859

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 n α n ∈ N where 0 < α < 1 , is the study of the sequence K 1 / l , where K is a polynomial having an l-th root in the field of formal power series. In this paper, we consider the sequence L ′ 1 / l , where L ′ is a polynomial having an l-th root in the field of formal power series and satisfying L ′ ≡ B mod C . Our main result is to prove the uniform distribution in the Laurent series case. Particularly, we prove the case for irreducible polynomials.

Highlights

In 1952, Carlitz [1] introduced the definition of equidistribution modulo 1 in the formal power series case which reveals profitable; it uses Weyl’s criterion [1], the generalisation of van der Corput inequality by Dijksma [2], and the theorem of Koksma by Mathan [3]
P describes the sequence of irreducible polynomials in F q[X]) with an l-th root (L(1/l)) (resp. (P(1/l))) in the field of formal power series
Madritsch and uswaldner in [6] called the maps f: F q[X] ⟶ G, where G is the group of Q-additives satisfying f(AQ + B) f(A) + f(B) for all polynomials A, B ∈ Fq[X] with deg(B) < deg(Q). ey proved the equidistribution of the sequence h(Wi), where h ∈ Fq(X− 1)[Y] is a polynomial with coefficients in the field of formal power series and (Wi) is an ordered sequence of polynomials in C(J) 􏼈A ∈ Ln: f(A) ≡ J mod M􏼉 if and only if one of the coefficients of h(Y) − h(0) is irrational

Summary

Introduction

In 1952, Carlitz [1] introduced the definition of equidistribution modulo 1 in the formal power series case which reveals profitable; it uses Weyl’s criterion [1], the generalisation of van der Corput inequality by Dijksma [2], and the theorem of Koksma by Mathan [3]. P describes the sequence of irreducible polynomials in F q[X]) with an l-th root (L(1/l)) In 2013, Mauduit and Car studied in [5] the Q− automaticity of the set of k-th power of polynomials in Fq[X] They calculated the number of polynomials K ∈ F q[X] with degree N such that the sum of digits of Kk in base Q is fixed. Ey proved the equidistribution of the sequence h(Wi), where h ∈ Fq(X− 1)[Y] is a polynomial with coefficients in the field of formal power series and (Wi) is an ordered sequence of polynomials in C(J) 􏼈A ∈ Ln: f(A) ≡ J mod M􏼉 if and only if one of the coefficients of h(Y) − h(0) is irrational. We will prove that the sequences (L{n′1/l}) and (P{n′1/l}) are equidistributed modulo 1

Preliminary

Proofs of Theorems 2 and 3

Results