Research Article Equidistribution Modulo 1

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.


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].
Car in [4], inspired by equidistribution modulo 1 of the sequence (n α ) n∈N where 0 < α < 1, characterised equidistribution modulo 1 of the sequences (L (1/l) ) and (P (1/l) ), where L describes the sequence of polynomials in F q [X] (resp. 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.
In 2013, Mauduit and Car studied in [5] the Q− automaticity of the set of k-th power of polynomials in F q [X]. Moreover, they calculated the number of polynomials K ∈ F q [X] with degree N such that the sum of digits of K k in base Q is fixed. In the same subject, Madritsch and uswaldner in [6] called the maps f: In this article, we are interested in the subsequences (L n ′ ) of (L n ) and (P n ′ ) of (P n ) of polynomials in arithmetic progression having an l-th root. We will prove that the sequences

Preliminary
Let F q be a finite field of characteristic p with q elements. We consider F q [X], F q (X), and F q ((X − 1 )) as analogues of Z, Q, and R, respectively.
with a i ∈ F q , n 0 ∈ Z, and a n 0 ≠ 0. We define ](f) � deg(f) � − n 0 and |f| � q deg(f) . We note [f] the polynomial part of f and f its fractionary part. Let Res(f) � a 1 if f ≠ 0, and sgn(f) � a n 0 . Let ψ: F q ⟶ C be a nontrivial additive character. For all f ∈ F q ((X − 1 )), we suppose that E(f) � ψ(Res(f)).
Let l be a positive integer >2 which is not divisible by the characteristic p of the field F q . We introduce L � a 1 , . . . , a r as the set of the r-th elements having an l-th root in F * q , and we have en, for f and g ∈ F * q ((X − 1 )), g is called an l-th root of f; we note f � g l if and only if ](f) ≡ 0 mod l and sgn(f) ∈ L. In particular, a nonzero polynomial A has an lth root in F * q ((X − 1 )) if and only if deg(A) ≡ 0 mod l and sgn(A) ∈ L.
We denote by L the set of polynomials with an l-th root in F * q ((X − 1 )): and if I is the set of irreducible polynomials over F q [X], we define P � L ∩ I. If n � S i�1 n i q i , where n i ∈ 0, . . . , q − 1 for all i ∈ 0, . . . , s { }, is the representation in base q of the integer n ≥ 1, then let H n � χ n 0 + · · · + χ n s X s , where χ n i are given by the bijection n i ↦χ n i from 0, . . . , q − 1 to F q . For n � 0 and 1, it is convenient to suppose that χ 0 � 0 and χ 1 � 1. We define the order in F q by χ n i < χ n i +1 , for all n i ∈ 0, . . . , q − 1 , and in F q by en, we order F q by posing for all natural numbers n: is paper is devoted to the study of equidistribution modulo 1 of a certain sequence in the field of Laurent formal power series. In 1952, Carlitz introduced and characterised equidistribution modulo 1 in the field of Laurent formal power series and obtained the following result.
Finally, we enounce a result which concerns a class of irreducible polynomials given by Artin in [7], which will be very useful later.
Theorem 1 (see [7]). Let C, B ∈ F q [X] be coprime polynomials. If π(m: C, B) denotes the number of monic irreducible polynomials with degree m which are congruent to B modulo C, then where θ is a constant <1. is theorem is analogous to the theorem of prime numbers in arithmetic progression.

Results
Let l ≥ 2 be an integer nondivisible by the characteristic p of the field F q ; we order the set of the l-th powers of L under the increasing order of F q , and we fix a polynomial C with degree c. For all B ∈ F q [X], we denote by L ′ the subset of L defined in (2), and P ′ the subset of P � L ∩ I: We ordered the elements of L ′ and P ′ with the order relation defined in (2); hence, e aim of this paper is to prove the following theorems.

Proofs of Theorems 2 and 3
4.1. Tools. A generalisation of eorem 1 was proved in 1965 by Hayes introducing the arithmetic progression.
Lemma 2 (see [8]). Let C ∈ F q [X] be a polynomial with degree c. en, for all polynomials B, there exist exactly q m− c monic polynomials with degree m which are congruent to B modulo C if m ≥ c.
Theorem 4 (see [8]). Let k ≥ 1 be a positive integer, u � (u 1 , . . . , u k ) be a sequence of k elements in F q , and C, B ∈ F q [X] be coprime polynomials. If, for m ≥ k, π(m; u, C, B) is the number of irreducible and monic poly- where θ is a constant <1.
e proofs of eorems 2 and 3 are based on Corollary 1 whose proof needs the following lemmas.
Let Z � A − K; we denote by π(lk: Y, ξ) the number of polynomials A ∈ A such that We obtain With the orthogonality criterion of ψ, it results in (ii) If A � J, then by eorem 4, we have We deduce that Finally, with the orthogonality criterion, we obtain where r is defined in (1). Let H ∈ F q [X] * with degree h, and N is an integer such that where [x] defines the least integer ≥x. e sequence (b m ) is strictly increasing, and there exists a unique integer t such that Moreover, there exists a unique integer s ∈ 0, . . . , r − 1, such that Let To prove eorem 2, we have to show that Using relations (24) and (25), we rewrite the sum W(N) to obtain with Journal of Mathematics (29) We start by giving an estimation of the sum W 1 which concerns the polynomials of L ′ with degree ≤l(t − 1). We have

(30)
We have to just major the first part of the sum by the number of polynomials with degree <((l(h + c))/(l − 1)), and we apply Corollary 1 on the second part to obtain W 1 ≤ q ((l(h+c))/(l− 1)) . (31) We apply the same Corollary 1 on W 2 that represents the sum of polynomials with degree lt and and with signature a 1 , . . . , a s− 1 , then (32) e polynomials in W 3 can be written in the form where j � 1 + b t− 1 + sa t and the sequence ( By the order relation on F q [X] (4), if and is the presentation in base q of the integer n N , we have To estimate W 3 , we will distinguish two cases: when the degree of H n N is up to the integer (l − 1)t − h − c − 1 and when it is not: 1st case: m ≤ (l − 1)t − h − c − 1. Using (34) and the fact that the sequence (n i ) j ≤ i ≤ N is strictly increasing, we obtain N − j ≤ n N − n j ≤ n N ≤ q m+1 − 1.
(36) us, 2nd case: m > (l − 1)t − h − c − 1. e polynomials H n i defined in (33) are of the form H n i � y 0 + y 1 X + · · · + y m X m ≤ H n N � χ c 0 + χ c 1 X + · · · + χ c m X m . (38) Let Y be the set of polynomials of the form such that, for every polynomial Z with degree If k is the greatest index i ∈ j, . . . , N for which L i ′ are written in the form with Y ∈ Y and Z being a polynomial with degree <(l − 1)t − h − c, then we rewrite the sum W 3 : with We have where π(ξ) is the number of couples where π(lt: Y, ξ) denotes the number of polynomials L ′ ∈ L such that with which gives the same arguments presented in the proof of Corollary 1, and then we deduce that In (34), let v be the least index >(l − 1)t − h − c such that c v ≠ 0; then, we have which coefficients satisfy the condition: is less then L n N ′ , we obtain which leads to en, with (48) and (53), it results in With (31), (32), and (54), we obtain and finally, with (24), we obtain which ends the proof.

Proof of eorem 3.
e proof of eorem 3 is treated as the proof of eorem 2, and we will keep the same notations with the appropriate modifications. Let π(m, C, B) be the number of monic irreducible polynomials with degree m in F q [X], congruent to B modulo C, satisfying with [4] the following property: In P ′ , we have a m � rπ(lm: C, B), and there exist constants c 1 > 0 and c 2 > 0 such that Let H be a nonzero polynomial with degree h and N be an integer satisfying (23). We suppose now that (59) With relations (24) and (25), we obtain By the same method used in the proof of eorem 2, with Corollary 1, we obtain and then where W 4 ′ is the sum defined in (42) concerning the polynomials in P ′ . Finally, we treat the sum With eorem 4, for all polynomials Y � a s X lt + y m X m + · · · + y v X v + · · · + y (l− 1)t− h− c X (l− 1)t− h− c , in which coefficients satisfy condition (51), there exists Irreducible polynomials P ′ are congruent to B modulo C such that deg(P ′ − Y) < (l − 1)t − h − c. Such polynomials P ′ are in P ′ , and we have n k ≥ c m q m + · · · + c v− 1 q v− 1 + · · · +(q − 1)q (l− 1)t− h− c n N − 2q (l− 1)t− h− c + 1.

Data Availability
No data were used to support this study.

Conflicts of Interest
e author declares that there are no conflicts of interest.