On recurrence in positive characteristic

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ScienceDirect Indagationes Mathematicae xx (xxxx) xxx–xxx www.elsevier.com/locate/indag

On recurrence in positive characteristic S. Kristensen a , A. Jaˇssˇov´a b , P. Lertchoosakul c , R. Nair b,∗

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a Department of Mathematics, Ny Munkegade 118, Building 1530, Room 4168000, Aarhus, Denmark b Mathematical Sciences, The University of Liverpool, Peach Street, Liverpool L69 7ZL, United Kingdom c Institute of Mathematics, Polish Academy of Sciences, ul. niadeckich 8, 00-656 Warszawa, Poland

Received 30 April 2014; received in revised form 17 November 2014; accepted 17 November 2014 Communicated by R. Tijdeman

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Abstract Let P − 1 denote the set of primes minus 1. A classical theorem of A S´ark˝ozy says that any set of natural numbers of positive density contains a pair of elements whose difference belongs to P − 1. An ergodic approach to questions of this type was given by the fourth author, building on work of H. Furstenberg. In this paper we give a proof of the positive characteristic analogue of this result using the same approach. c 2014 Published by Elsevier B.V. on behalf of Royal Dutch Mathematical Society (KWG). ⃝

Keywords: Fields of formal power series; Positive characteristic; Invariant measures; Poincar´e recurrence; Intersectivity

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1. Introduction Let q be a power of a prime p and Fq be the finite field with q elements. Denote by Fq [t] and Fq (t) the ring of polynomials with coefficients in Fq and the quotient field of Fq [t] respectively. For each P/Q ∈ Fq (t) set |P/Q| = q deg(P)−deg(Q) where for an element g ∈ F p [t] we have denoted its degree by deg(g). Note that with respect to the valuation function | · | the integral domain Fq [t] is a Euclidean domain and so also a principal ideal domain and a unique factorization domain. In particular Bezout’s identity is satisfied. Let Fq ((t −1 )) denote the field of formal ∗ Corresponding author.

E-mail addresses: [email protected] (S. Kristensen), [email protected] (A. Jaˇssˇov´a), [email protected] (P. Lertchoosakul), [email protected] (R. Nair). http://dx.doi.org/10.1016/j.indag.2014.11.003 c 2014 Published by Elsevier B.V. on behalf of Royal Dutch Mathematical Society (KWG). 0019-3577/⃝

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Laurent series   Fq ((t −1 )) = an X n + · · · + a0 + a−1 X −1 + · · · : n ∈ Z, ai ∈ Fq . Elements of Fq ((t −1 )) that are not in Fq (t) will be referred to as irrational. Also dq (x, y) = |x − y| for x, y ∈ Fq (t) defines a metric on Fq (t). The metric extends to Fq ((t −1 )) by completion and by implication to its subset L = {x ∈ Fq ((t −1 )) : |x| ≤ 1}. Note that this metric is non-Archimedean since |x + y| ≤ max(|x|, |y|). As Fq [t] is a unique factorization domain, each of its elements has a canonical factorization into a unique product of irreducible elements—in effect the fundamental theorem of arithmetic for the ring Fq [t]. Because of this we can speak of two elements P, Q of Fq [t] as being coprime in a well defined sense. We write this briefly as (P, Q) = 1. As usual we say a polynomial p(x) defined over Fq [t] is irreducible if it cannot be expressed as a product of two non-constant polynomials of degree less than that of p(x). We can think of the irreducible elements of Fq [t] as analogues of the prime numbers in Fq [t]. Chapter 1 of [16] gives more general background to the study of the field Fq [t]. Suppose M and N are positive integers with M > N . Let I (N , M) denote the ‘interval’ {g ∈ Fq [t] : N < deg(g) ≤ M} in Fq [t] and denote its cardinality which is q M+1 − q N +1 by |I (M, N )|. Let I = (In )n≥0 denote a sequence of such intervals whose lengths tend to infinity as n does, and let b(E, I ) = lim sup n→∞

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|E ∩ In | . |In |

We define the ‘Banach Density’ of E to be b(E) = sup b(E, I ). I

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Here the supremum is taken over all such sequences of intervals I = (In )n≥0 . We will call a subset G of Fq [t] intersective if given any S ⊆ Fq [t] with b(S) > 0 there exists x, y ∈ S such that x − y ∈ G. Our main theorem is the following. Theorem 1. Let P denote the set of irreducible polynomials in Fq [t] with leading unit coefficient. Then P − 1 = { p − 1 : p ∈ P} is intersective. This result is the analogue of a result due to A. S´ark˝ozy [15] where the role of Fq [t] is played by Z. The analogous result with P −1 replaced by P +1 follows in a very similar fashion. Unlike S´ark˝ozy, who only used analytic number theory, following ideas of H. Furstenberg [6,7], we use ergodic theory as well. In [12] we classified which polynomials φ are such that {φ( p) : p ∈ Π } where Π denotes the set of prime numbers. In fact the result was proved by the fourth author in the late 1985. See also [5] for some subsequent developments. Proving an analogue of this result from [12] on Fq [t] would require an improved understanding of uniform distribution of polynomials in irreducibles on L. Let (X, β, µ) be a probability space, with set X , σ -algebra β and measure µ. Suppose T : X → X is a measurable and measure preserving transformation i.e. T −1 B ∈ β if B ∈ β and µ(T −1 B) = µ(B) for all B ∈ β. An action of a monoid M on (X, β, µ) is a family of measurable measure preserving maps (Tm )m∈M of (X, β, µ) such that Tm 1 +m 2 (x) = Tm 1 ◦ Tm 2 (x) for all m 1 , m 2 ∈ M and all x ∈ X .

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We say a sequence G = (gn )n≥0 in the Fq [t] is recurrent if given any action (X, β, µ, (Tm )m∈F ) of Fq [t] and any A ∈ β with µ(A) > 0 there exists g ∈ G such that µ(A∩Tg−1 A) > 0. The following lemma gives the link between intersectivity and recurrence. In the setting of Z, this is called the Furstenberg transference principle. In fact, as shown by A. Bertrand-Mathis [1], intersectivity and recurrence are equivalent on Z.

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Lemma 1. A set G ⊆ Fq [t] is intersective if it is recurrent.

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Proof. Let Ω = {0, 1}Fq [t] . Clearly a point in Ω can be seen as the indicator function of a specific sequence contained in Fq [t]. We endow Ω with a metric by setting   d(x, y) = inf k −1 : x(k) ̸= y(k) .

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Here we have represented arbitrary elements x and y in Ω as sequences (x(k))k≥0 and (y(k))k≥0 , where the order of the terms in the sequence is any one compatible with increasing degree in Fq [t]. This gives rise to a topology that coincides with the Tychonoff product topology on Ω . Let (gn )n≥1 denote the sequence of elements in Fq [t] indicated by the element x ∈ Ω . We define an action (Tg )g∈Fq [t] of Fq [t] on Ω by setting Tg x to be the indicator function of the sequence (gn + g)n≥1 . Let η = I S denote the indicator function of the set S ⊆ Fq [t], which we assume to have positive Banach density. This may be viewed as a point in Ω in the obvious way. Let X denote the closure of {Tg η}g∈Fq [t] and let A = {ω ∈ X : ω(0) = 1}. There is a measure µ on X such that µ(A) > 0. This is because if we set 1  δT η , (k = 1, 2, . . .) µk = |Ik | n∈I n

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(where δx denotes the measure X supported at x) and (Ik )k≥0 is a collection of intervals with |Ik | → ∞ as k → ∞ such that lim

k→∞

|Ik ∩ S| = B(S) > 0. |Ik |

The weak star limit of the sequences {µk }k≥1 , which by its construction evidently preserves the elements of {Tg }g∈Fq [t] , satisfies µ(A) = lim µk (A) = k→∞

|Ik′ ∩ S| = B(S) > 0. |Ik |

Here (Ik′ )k≥1 is some subsequence of (Ik )k≥1 . By hypothesis, there is g ∈ G such that µ(A ∩ Tg−1 A) > 0. In particular there is an ω ∈ A such that ω(0) = ω(k) = 1, where k is the place the sequence in Fq [t] corresponding to g. Because ω belongs to the closure of (Tg η)g∈Fq [t] . This means there exist h 0 ∈ Fq [t] such that Th 0 η(0) = Th 0 η(k) = 1. Thus both the elements of Fq [t]. Corresponding to both h 0 and h 0 + k both belong to S as required.  For x ∈ Fq ((t)) let [x] denote the unique element P ∈ Fq [t] such that deg(x − P) < 0. Now let {x} = x − [x]. We call a−1 the residue of the element x = an X n + · · · + a0 + a−1 X −1 + · · · contained in Fq [t] and denote it by r es(x). Note that Fq is a finite dimensional vector space over F p . To each element α of Fq we can associate a map m α : Fq → Fq , defined for x ∈ Fq by m α (x) = αx. This is clearly a linear map in x. Over F p the map m α is represented by a finite dimensional matrix. We use tr(α) to denote the trace of this matrix, in the ordinary linear algebra sense of the term. This concept can be seen to be well defined in the sense that tr(α) can be

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checked to be independent of the choice of basis of Fq over F p . Let tr(.) denote the trace map from Fq to F p . Then there is a non-trivial additive character eq : Fq [t] → C× defined by setting eq (x) = e(tr(x)), where F p is embedded as a subgroup of the circle group S 1 ∼ = [0, 1) in the obvious way and for a real x we have as usual e(x) := e2πi x . We say a sequence of complex numbers (xn )n∈Fq [t] is positive definite if given any other sequence (z n )n∈Fq [t] , only a finite number of whose terms are non-zero we have  xn−m z n z m ≥ 0. n,m∈Fq [t]

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Suppose ⟨, ⟩ denotes the standard inner product on the Hilbert space L 2 (X, β, µ) and suppose ∥·∥ denotes the norm on L 2 (X, β, µ). Let xn = ⟨ f, Tn f ⟩(n ∈ Fq [t]). Then given any (z n )n∈Fq [t] , only a finite number of whose terms are non-zero we have  2        xn−m z n z m =  xn z n   ≥ 0. n∈Fq [t]  n,m∈Fq [t] Thus (xn )n∈Fq [t] is positive definite. We have the following implication of the Bochner–Herglotz characterization of positive definite functions on a locally compact Abelian group. See p. 22 of [14] for details. Lemma 2. Suppose f ∈ L 2 (X, β, µ), then there is a Borel measure ω f on L such that  ⟨ f, Tn f ⟩ = e(nα)dω f (α) (n ∈ Fq [t]). L

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Let U be a strongly continuous unitary representation of a locally compact Abelian group G on a Hilbert space H . If µ is a probability measure on G then      Ug , h dµ(g). Ug f dµ, h = A sequence of probability measures (µn )n≥1 is called a generalized summing sequence if for every unitary representation of U on H and every f ∈ H , the sequence ( Ug f dµn )n≥1 converges in mean to an invariant element of H . We have the following theorem [2]. Theorem 2. The following two statements are equivalent. (i) The sequence (µn )n≥1 of probability measures on the locally compact Abelian group G is a generalized summing sequence; and (ii) for each character χ on G not equal to the identity the Fourier transform µn (χ ) converges to 0 as n → ∞.  Let µ N = |G1N | n∈G N δn (N ≥ 1), where δn is the delta measure at n in Fq [t] and G N denotesthe set of elements from Fq [t], that are of absolute value no greater than N . Let N if ord({x}) < −N and that FN (x) = n∈G N e(nx). In [10] it is shown that FN (x) = q FN (x) = 0 if ord({x}) ≥ −N . From this it is immediate using Theorem 2 that (µ N ) N ≥1 is a generalized summing sequence on Fq [t]. From this we obtain the following analogue of the Von Neumann mean ergodic theorem on Fq [t]. Theorem 3. Let (X, β, µ, (Tg )g∈Fq [t] ) denote a measurable, measure preserving action of Fq [t] on the set X . Suppose f ∈ L 2 (X, β, µ). Then there is a function f invariant under (Tg )g∈Fq [t] ,

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such that

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    1    f (Tn x) − f (x) = 0. lim   N →∞  |G N | n∈G N

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Suppose G is a compact topological group. Recall that we say (xn )n≥1 ⊂ G is uniformly distributed on G, if for each continuous function f : G → C we have  N 1  lim f (xn ) = f (t)dt. N →∞ N G n=1 A tool central to our progress is the following version of Weyl’s criterion for uniform distribution [3]. Theorem 4. A sequence (xn )n∈Fq [t] ⊂ Fq ((t −1 )) is uniformly distributed on L if and only if for any m ∈ Fq [t] \ {0}, we have  1 e(mxn ) = 0. lim N →∞ |G N | x ∈G n

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We digress briefly to consider some arithmetical issues in Fq [t], which may be well known but a convenient reference to which is unknown to the authors. Given c ∈ Fq [t] with |c| > 1 let Fq [t]/cFq [t] denote the collection of cosets r + cFq [t] with |r | < |c|. Because the analogue of Bezout’s theorem holds in Fq [t] if (s, t) = 1 then there exist x, y ∈ Fq [t]/cFq [t] such that sx + t y = 1. In particular if (u, c) = 1 then there is a unique and well defined solution to the equation ux ≡ 1(mod c) in Fq [t]/cFq [t]. Here of course ≡ denotes belonging to the same coset in Fq [t]/cFq [t]. We will use G q,c to denote the set of reduced ‘residue classes’ in Fq [t]/cFq [t]. Using right cancellation on the multiplicative group G q,c , given d ∈ G q,s we have Πx∈G q,c (d x) ≡ Πx∈G q,c x (mod c). If we let φq (c) be the cardinality of the finite group G q,c . We have an analogue of the Fermat–Euler theorem d φq (c) ≡ 1(mod c). Theorem 5. Let Pe, f , with |e| > 1, | f | > 1, denote the set of elements of P congruent to e modulo f . Then for irrational x ∈ Fq ((t −1 )) the sequences ({ px}) p∈Pe,q are uniformly distributed in L. Proof. The proof relies on adapting an idea from [17]. The result that the sequence ({ px}) p∈P is uniformly distributed in L for irrational x, appears in [13]. Our proof adapts this fact using basic facts about Dirichlet characters on Fq [t] which we recap briefly. See [13,8] for details. By a Dirichlet character on Fq [t] we mean a function χ : Fq [t] → C× such that χ (x y) = χ (x)χ (y) and χ (x + m) = χ (x) for all x, y ∈ Fq [t] and some m ∈ Fq [t]. We call m the conductor or modulo of χ. One checks easily that χ (1) = 1 and if χ is not identically one, then χ (0) = 0. Define χ0 , which we call the principal character by χ0 (n) = 1 if (n, m) = 1 and χ0 (n) = 0 otherwise. Using the group structure of G we obtain the orthogonality relations: (1) q,c n mod m χ (n) =   φq (m) if χ = χ0 and χ (n) = 0 if χ = ̸ χ and (2) χ (n) = φq (m) if 0 n mod m n mod m  n ≡ 1 mod m and n mod m χ (n) = 0 otherwise. Evidently residues modulo N give rise to residues modulo M if M divides N (by discarding information). The effect on characters is in the opposite direction. A character χ modulo M, induces one modulo N for a multiple N of M. We call a character primitive if it is not induced from of a proper divisor. Given a character χ

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 modulo m, we can attach to it a ‘Ramanujan sum’ cχ (n) = b mod m χ (b)e( mb n). In the special case n = 1 this reduces to the ‘Gauss sum’ τm (χ ), which satisfies |τm (χ )|2 = |Fq [t]/mFq [t]| = |m|. In light of Theorem 4 and the analogue of Dirichlet’s theorem on primes in arithmetic progressions on Fq [t] [13], proving Theorem 5 reduces to showing  1 e(npx) = 0, lim N →∞ π N , f,e p∈G N p≡f e

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where π N , f,e denotes the number of elements of P, congruent to e mod f contained in G N , and n is an arbitrary non-zero element of Fq [t]. Here we have written n ≡b a to denote n is congruent to a mod b. Note using the orthogonality relations, that if      1     S :=  e(npx)  π N , f,e p∈G  N   p≡f e         1 1  e(npx) =  χ ( p)χ (e) . φq (e)   π N ,q,e p∈G ˆ χ ∈G f,e

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Here Gˆ f,e denotes the set of Dirichlet characters modulo f . This means that there is a character χ ∈ Gˆ f,e such that    1     S≤ e(npx)χ ( p) .  π N , f,e p∈G  N

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We consider two cases. Case 1: First assume χ is primitive. Then       1   S≤ e(npx) .  π N . f,e p∈G N ;( p,q)=1  Hence    1     S≤ e(npx) + f.   π N , f,e p∈G N

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Case 2: Now assume χ is imprimitive and induced by χ1 mod f 1 with f 1 dividing f . Recall from our earlier discussion, that the Gauss sum corresponding to the primitive character χ1 modulo f 1 is defined as    a τ f1 (χ ) = χ1 (a)e . f1 a∈F / f F q

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Thus, taking complex conjugates using left cancellation on the group Fq / f 1 Fq .    pa τ f1 (χ1 ) = χ 1 ( p) χ 1 (a)e − . f1 a∈F / f F q

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Also recall that |τ1 (χ1 )|2 = | f 1 |. This means that if χ is induced by χ1 then,   τ f1 (χ1 )  |τ f1 (χ1 )|2 pa . = χ (a)e − χ ( p) = χ1 ( p) | f1| | f 1 | a∈F / f F 1 f1 q

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Thus there exists C1 > 0 such that         1  τ (χ ) pa f 1  + C1 . S ≤  e(npx) 1 χ 1 (a)e − | f 1 | a∈F / f F f 1  π N , f,e  π N , f,e p∈G ;( p, f )=1 q

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So there exists C2 > 0 such that         1  τ (χ ) pa f 1  + C2 S ≤  χ 1 (a)e − e(npx) 1 | f 1 | a∈F / f F f 1  π N , f,e  π N , f,e p∈G N q q    1   pb  C2  e npx − . ≤ +  π N ,q,e p∈G q  π N , f,e

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Combining cases 1 and 2, Theorem 5 is proved in light of the uniform distribution of ( pn x)n≥1 in L for irrational x.  Proof of Theorem 1. A N f (x) =

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then

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H N (x)dω f (x)

(N ≥ 1).

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Also lim N ≥1 H N (x) = 0 if x ̸= 0 and H N (0) = 1 for large N so

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where by ω f (0) we mean, any atom of ω f sitting at 0. On the other hand, using Theorem 3, we see that   f, f = lim ⟨ f, A N f ⟩ .

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Also as f is (Tg )g∈Fq [t] invariant,     f, f = Tg f, f , (g ∈ Fq [t])   = A N f, f (N ≥ 1)   = f, f .

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Thus applying Cauchy’s inequality 2  2        ω f (0) = f , f ≥  f dµ =  f dµ . L

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Let L s (s = 1, 2, . . .) denote the subset of P − 1 that is divisible by the least common multiple of the elements of G s \ {0}. Also let L s,N = L s ∩ G N (N ≥ 1). and let a  Fk = : 1 ≤ |a| < |b| ≤ k , (k = 1, 2, . . .) b c and let Fk denote the complement of Fk in Fq (t) ∩ L. Also let ω f = ωi + ωr , where ωr denotes the atoms of the measure ω f on Fq (t) ∩ L. We then have for n ∈ Fq [t]        a     na  ⟨ f, Tn f ⟩ = ωr e , e + e(nx)dωi (x) + ω f (0) +   b q L a a ∈F ∈F c b

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    a    na  1  . e ωr e b |L k0 ,N | n∈L b

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Ms,N ,r = { p − 1 : p ≡ r mod s ∗ ; p ∈ Iq } ∩ G N , where gs ∗ denotes the non-empty set of reduced residues mod s ∗ such that r ≡ 1 mod s ∗ . Thus    e(nx) = e(nx), r ∈gs ∗ n∈Mk ,N ,r 0

which using the fact that ({( p − 1)x}) 

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Let s ∗ denote the least common multiple of the elements in G s . Also let Iq denote the set of irreducibles in Fq [t]. Now let

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where k0 = k0 (ϵ) is chosen so the second term on the right is less than ϵ. Thus      1 1  ⟨ f, Tn f ⟩ = e(nx) dω f (x) c |L k0 ,N | n∈L |L k0 ,N | n∈L L∩(F [t]) q k0 ,N k0 ,N   a   ωr e + ωr ({0}) + b a ∈F b

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|Mk+0,N ,r | = o(|L k0 ,N |).

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Thus the right hand side of (2) tends to zero as N tends to ∞. Also the expression in (3) expression is <ϵ in absolute value. Hence if we take f = χ B for B ∈ β with µ(B) > 0 by (1)  1 µ(B ∩ Tn−1 B) ≥ µ(B)2 − ϵ. lim sup N →∞ |L k0 ,N | n∈L k0 ,N

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Thus P − 1 is both intersective and recurrent. In [4] it is shown that if T (x) is a polynomial of degree less than p, with coefficients in Fq ((t −1 )), at least one of which, other than the constant term, is irrational, then (T (x))x∈Fq [t] is uniformly distributed on L. Using a variant of the above method, it is also possible to show that if S is a polynomial of degree less than p with coefficients in Fq [t], then assuming S(0) = 0, the

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sequence (S(x))x∈Fq [t] is recurrent. See also [11] for a recent extension of this result. In [11] the authors are able to get around the requirement that the degree of T is less than p under certain circumstances. The method in [4] relies on exponential sums and the restriction to the degree of T being less than p arises from the method of Weyl differencing. The result in [11] is proved by avoiding Weyl differencing and using a large sieve argument instead. 

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Uncited references

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[9].

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References [1] A. Bertrand-Mathis, Ensembles intersectifs et r´ecurrence de Poincar´e [Intersective sets and Poincar´e recurrence], Israel J. Math. 55 (2) (1986) 184–198 (in French). [2] J. Blum, B. Eisenberg, Generalized summing sequences and the mean ergodic theorem, Proc. Amer. Math. Soc. 42 (1974) 423–429. [3] L. Carlitz, Diophantine approximation in fields of characteristic p, Trans. Amer. Math. Soc. 72 (1952) 187–208. [4] A. Dijksma, Uniform distribution of polynomials over GFq, x in GF[q, x] II, Nederl. Akad. Wetensch. Proc. Ser. A Indag. Math. 32 (1970) 187–195. [5] N. Frantzikinakis, B. Host, B. Kra, The polynomial multidimensional Szemer´edi theorem along shifted primes, Israel J. Math. 194 (1) (2013) 331–348 (English summary). [6] H. Furstenberg, Ergodic behavior of diagonal measures and a theorem of Szemer´edi on arithmetic progressions, J. Anal. Math. 31 (1977) 204–256. [7] H. Furstenberg, Recurrence in Ergodic Theory and Combinatorial Number Theory, Princeton University Press, 1981. [8] D.R. Hayes, The distribution of irreducibles in GF[q, x], Trans. Amer. Math. Soc. 117 (1965) 101–127. [9] T. Kamae, M. Mend`es France, van der Corput’s difference theorem, Israel J. Math. 31 (3–4) (1978) 335–342. [10] H.M. Kubota, Waring’s problem problem for F p [x], Dissertationes Math. (Rozprawy Mat.) 117 (1974) 60. [11] T.H. Le, Y.-R. Liu, Equidistribution of polynomial sequences in function fields, with applications, arXiv:1311.0892 [math.NT]. [12] R. Nair, On certain solutions of the Diophantine equation x − y = p(z), Acta Arith. 62 (1) (1992) 61–71. [13] G. Rhin, R´epartition modulo 1 dans un corps de s´eries formelles sur un corps fini, Dissertationes Math. (Rozprawy Mat.) 95 (1972) 75 (in French). [14] W. Rudin, Fourier Analysis on Groups, in: Interscience Tracts in Pure and Applied Mathematics, vol. 12, John Wiley and Sons, 1962. [15] A. S´ark˝ozy, On difference sets of sequences of integers, II, Ann. Univ. Sci. Budapest. E˝otv˝os Sect. Math. 21 (1978) 45–53. [16] M.H. Taibleson, Fourier Analysis on Local Fields, in: Mathematical Notes, vol. 15, Princeton University Press, 1975. [17] M.A. Wodzak, Primes in arithmetic progression and uniform distribution, Proc. Amer. Math. Soc. 122 (1) (1994) 313–315.

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