INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 5 (2005), #A04

ON THE DISTRIBUTION OF DISTANCES BETWEEN THE POINTS OF AFFINE CURVES OVER FINITE FIELDS

B.V. Petrenko

Department of Mathematics, Texas A&M University, College Station, Texas 77843-3368, USA

                           petrenko@math.tamu.edu

Received: 9/23/04, Accepted: 2/5/05, Published: 2/15/05

Abstract

Let 𝔽_{q} be a finite field with q elements, 𝔽̄_{q} an algebraic closure of 𝔽_{q}, and 𝔸^{n} \binom{}{𝔽̄_{q}} an ndimensional affine space over 𝔽̄_{q}. Let C be an affine absolutely irreducible curve in 𝔸^{n} \binom{}{𝔽̄_{q}}. We interpret the points of C over 𝔽_{q} as points in the cube [−1, 1]^{n−1}. The main result of this paper is an asymptotic formula for the distribution of points of C in [−1, 1]^{n−1} provided the characteristic p of 𝔽_{q} is large, while n, log_{p} q are fixed, and the degree of C is bounded. When p = q, this becomes a recent result of Cobeli and Zaharescu.

Keywords: finite field, curves over finite fields, distribution of points, Bombieri’s inequality, principle of Lipschitz, Weil’s theorem.

AMS Subject Classification: 11G20, 11L05, 11L07, 11T23.

1. Introduction

This paper gives a generalization the main result of Cobeli and Zaharescu [3]. The main result of [3] generalizes that of Zheng [7] and partially generalizes that of Zhang [6]. This will be explained in detail below.

Let p be prime number, and q = p^{m}. Let C be a curve of degree d in an affine space 𝔸^{r}(𝔽̄_{q}), where 𝔽̄_{q} is an algebraic closure of a finite field with q elements 𝔽_{q}. The goal of this paper is to study the distribution of the points of C in the cube [−1, 1]^{mr−1} ⊆ ℝ^{mr−1}.

Let us begin by explaining how the points of C are interpreted as points of [−1, 1]^{mr−1}. The resulting set will be called Ñ_{C}. We assume that C is not contained in a hyperplane ^{1} of 𝔸^{mr}(𝔽̄_{p}). There is a bijection between 𝔽_{p} = ℤ/pℤ and {0, …, p − 1} via l + pℤ ↦ l. Then 𝔽_{q} can be identified with {0, …, p − 1}^{m}, and it makes sense to consider the set

1In this paper, all hyperplanes are assumed to be affine, i.e. given by an equation ∑_{k=1}^{mr} α_{k}x_{k} = c for some c, α_{1}, …, α_{mr} ∈ 𝔽_{p}.

N_{C} = {x = (x_{1}, …, x_{mr}) ∈ {0, …, p − 1}^{mr} | x ∈ C}.  (1)

In general, each such an identification of C with N_{C} corresponds to a basis of 𝔽_{q} over 𝔽_{p}. Nevertheless, the main result of this paper, Theorem 1, is independent of this choice.

Sometimes we will prefer to think of N_{C} as a subset of 𝔽_{p}^{mr}. This is legitimate because N_{C} will be regarded as the domain of some p-periodic functions related to the exponential function e^{\frac{2πix}{p}}.

Definition 1 We will write e(t) instead of e^{2πit}.

Next we consider the map

: ℝ^{mr} → ℝ^{mr−1}, x = (x_{1}, …, x_{mr}) ↦ x̃ = (x̃_{1}, …, { \atop x_{mr−1}}),  (2)

where

x̃_{j} = \frac{x_{j+1} − x_{j}}{p} .  (3)

In this paper we obtain some asymptotic results about the the set

Ñ_{C}  (4)

endowed with the probability measure μ_{C} defined by the formula

μ_{C}(Ω) = \frac{#\{x ∈ N_{C} | x̃ ∈ Ω\}}{#N_{C}}.  (5)

By placing some restrictions on C, the denominator of (5) may be estimated by a well known theorem of A. Weil. Therefore, the point of this paper is to estimate the numerator of (5). This will involve the theorems of Bombieri ([1], Th. 6, p. 97), Davenport [4], and Weil. Our proof extends that of Cobeli and Zaharescu [3].

Before formulating the main result of this paper, Theorem 1, we would like to describe a certain class D_{n}(h) of subsets in ℝ^{n}. (A more general class has been introduced in an important paper of Davenport [4].) Our Definition 3 below is more restrictive than Davenport’s because we additionally impose Conditions 4 and 5.

Remark 1 It would be desirable not only to relax these restrictions in Definition 3, but also to generalize the main result of [4].

To make the formulation of Definition 3 less cumbersome, we introduce the following auxiliary Definition 2 Let X ⊆ ℝ^{n} and τ > 0. We introduce the following tessellation of ℝ^{n}:

ℝ^{n} = ⋃_{t∈ℤ^{n}}(1/τ)t + [0, 1/τ]^{n}.  (6)

1. We define the sets I_{τ}(X) and E_{τ}(X) to be the unions of cubes in (6) contained in X and intersecting X, respectively.

2. For any ε > 0, we define the set E_{τ,ε} (X) = X ∪ Δ_{τ,ε} , where Δ_{τ,ε} consists of the points of E_{τ} (X) whose standard distance to the boundary of E_{τ} (X) is at least ε.

Now we are ready to give the following

Definition 3 Let X ⊆ ℝ^{n} and h a positive integer. Then X ∈ D_{n}(h) if all of the following conditions are satisfied:

1. X is compact.

2. Any line parallel to one of the n coordinate axes intersects X in at most h intervals.

3. The same is true for any m ∈ {1, …, n − 1} and any projection of X on any of the m-dimensional coordinate subspaces.

4. Let V (·) denote the (n-dimensional) volume of the set. Then V (X) exists and has the following properties: V (X)−V (I_{τ}(X)) = O_{X}(1/τ) and V (X)−V (E_{τ}(X)) = O_{X}(1/τ) as τ → +∞.

5. There exists τ_{0} ≥ 1, depending on X, such that for any τ ≥ τ_{0},

(a) All but O_{X}(τ^{n−1}) vertices of the grid (6) in I_{τ}(X) possess the following property. For a vertex v there exists a cube C_{v} of (6) with v ∈ C_{v} ⊆ X.

(b) Any vertex w ∈ E_{τ}(X)⧹X of the grid (6) is a boundary point of E_{τ}(X).

(c) The sets I_{τ}(X), E_{τ}(X), and E_{τ,ε} (X), for any ε > 0, satisfy Conditions 2 and 3 of this definition.

In some situations, the values of h or n are irrelevant. Therefore, we consecutively define the following two classes of sets:

Definition 4 D_{n} = ⋃_{h=1}^{∞} D_{n}(h).

Definition 5 D = ⋃_{n=1}^{∞} D_{n}.

At this point, we would like to recall the main result of Davenport [4]. We state it for a possibly smaller class of sets, as explained above.

Remark 2 Let X ∈ D_{n}(h), N(X) the number of points with integral coordinates in X, V (X) the n-dimensional volume of X, V_{j}(X) the sum of the j-dimensional volumes of the projections of X on all the j-dimensional coordinate subspaces, and V_{0}(X) = 1 by definition. Then

|N(X) − V (X)| ≤ ∑_{j=0}^{n−1} h^{n−j}V_{j}(X).  (7)

Next we define the function g_{n}. Its support is a polytope contained in the cube [−1, 1]^{n}. The function g_{n} will be used to define the probability measure μ_{C} in Theorem 1.

Definition 6

g_{n}(t_{1}, …, t_{n})  =  max {0, min_{1≤k≤n} {1, 1 − ∑_{s=1}^{k} t_{s}} + min_{1≤k≤n} {0, ∑_{s=1}^{k} t_{s}}} . (8)

The main result of this paper is the following

Theorem 1 Let {q_{j}} an increasing sequence of powers of primes {p_{j}} with log_{p_{j}} q_{j} = m = const. Let C_{j} be an irreducible affine algebraic curve in 𝔸^{r}(𝔽̄_{q_{j}}) of degree ≤ d = const. Suppose that C_{j} is not contained in a hyperplane of 𝔸^{mr}(𝔽̄_{p_{j}}). Then for any Ω ∈ D_{mr−1},

μ_{C_{j}}(Ω) = ∫_{Ω} g_{mr−1}(t)dt + O_{m,r,d,Ω} (q_{j}^{− \frac{1}{2(mr+1)}} ln^{\frac{mr}{mr+1}} q_{j})  (9)

as j → +∞. (Here ln(·) denotes the logarithm to base e.)

In other words, the measures μ_{C_{j}} weakly converge, as j → +∞, to μ_{mr−1} with density function g_{mr−1}. We believe that Theorem 1 is of interest because g_{mr−1} is independent of d, {q_{j}}, {C_{j}}.

Corollary 2 If m = 1, then Theorem 1 becomes the main result of Cobeli and Zaharescu [3]. In turn, let f ∈ ℤ[x, y] be of degree d ≥ 2, and suppose that f is absolutely irreducible modulo all large primes. If C_{j} is a plane curve obtained by reducing f modulo p_{j}, then we obtain the main result of Zheng [7]. Finally, if the curve is of the form f(x, y) = xy − 1, then we partially recover the main result of Zhang [6].

Remark 3 Theorem 1 may be strengthened as follows: m and r may be allowed to depend on the curve, provided they are bounded. This formulation of Theorem 1 leaves only finitely many possibilities for the values of m and r, and therefore the same proof is valid in this situation as well.

Remark 4 It may be of interest to prove that the formula

ν_{C}(Ω) = \frac{#Ñ_{C} ∩ Ω}{#Ñ_{C}}

defines an asymptotically well defined conditional probability measure on ℝ^{mr−1}, and to compare its asymptotic behavior to that of μ_{C}.

Remark 5 By Weil’s theorem, the number of points in the projective closure of the curve C_{j} is q_{j} + O_{m,r,d} (q_{j}^{1/2}). By the assumptions in Theorem 1, however, the same estimate holds for the curve C_{j} itself. A similar observation about the more general Lang-Weil estimates of [5] has been made in [2], p. 120. This observation also allows us to apply the result of Bombieri ([1], Th. 6, p. 97).

Acknowledgments. I thank Alexandru Zaharescu for suggesting the problem to me and for his generous sharing of ideas with me. I thank Nigel Boston for many very helpful discussions and for his wonderful hospitality. Anand Pilay has kindly pointed my attention to the paper of Z. Chatzidakis, L. van den Dries, and A. Macintyre [2]. Jeremy Tyson has very helpfully commented on the definition of D_{n}(h).

2. Proof of Theorem 1

To simplify the notation, we will denote p_{j}, q_{j}, C_{j} by p, q, C, respectively.

2.1 The map ^{∗}.

If (x_{1}, …, x_{mr}) ∈ ℝ^{mr}, then define

y = \frac{x_{1}}{p} , t_{1} = \frac{x_{2} − x_{1}}{p} , …, t_{mr−1} = \frac{x_{mr} − x_{mr−1}}{p} .  (10)

Let (x_{1}, …, x_{mr}) ∈ N_{C}, then 0 ≤ x_{1}, …, x_{mr} ≤ p − 1. Therefore

0 ≤ y, y + ∑_{j=1}^{k} t_{j} = \frac{x_{k+1}}{p} ≤ 1, where k ∈ \{1, …, mr − 1\}.  (11)

This can be restated as follows:

0 ≤ y ≤ 1, − ∑_{j=1}^{k} t_{j} ≤ y ≤ 1 − ∑_{j=1}^{k} t_{j}, k ∈ {1, …, mr − 1}.  (12)

Based on these considerations, for Ω ∈ D_{mr−1}, we define the set Ω^{∗} by

Ω^{∗} = {(y, t_{1}, …, t_{mr−1}) ∈ ℝ × Ω | y, t_{1}, …, t_{mr−1} satisfy (12)}.  (13)

We remark that Ω^{∗} ∈ D_{mr}.

From (13), we see that the set Ω^{∗} can be described as a cylinder bounded by some two hypersurfaces, h(t_{1}, …, t_{mr−1}) and H(t_{1}, …, t_{mr−1}), as follows:

Ω^{∗} = {(y, t) ∈ ℝ^{mr} | 0 ≤ y ≤ 1, t ∈ Ω, h(t) ≤ H(t)} .  (14)

We proceed to find equations of these hypersurfaces based on (12).

1. For a given t ∈ Ω, the smallest value of y is

h(t_{1}, …, t_{mr−1}) = max{0, −t_{1}, …, −t_{1}, + … + t_{mr−1}} = − min{0, t_{1}, …, t_{1} + … + t_{mr−1}}. (15)

2. For a given t ∈ Ω, the largest value of y is

H(t_{1}, …, t_{mr−1}) = min{1, 1 − t_{1}, …, 1 − t_{1} − … − t_{mr−1}}.  (16)

We see that

g_{mr−1}(t) = max {0, H(t) − h(t)} .  (17)

Therefore, the volume of Ω^{∗} may be written as follows:

V(Ω^{∗}) = ∫_{Ω} g_{mr−1}(t_{1}, …, t_{mr−1})dt_{1} … dt_{mr−1}.  (18)

We will use this formula in (54) of 2.4.3.

2.2 Main idea

We will use several rescalings below. One could think of them as changing the unit of length. We firstly pass from Ω^{∗} to pΩ^{∗}, because there is a bijection between N_{C} and the set of integral points of pΩ^{∗}, namely,

(x_{1}, …, x_{mr}) = x ↦ px^{∗} = (x_{1}, x_{2} − x_{1}, …, x_{mr} − x_{mr−1}) .  (19)

In particular, we may express the numerator of (5) as follows:

#{x ∈ N_{C} | x̃ ∈ Ω} = N (preimage of Ω under ) = N(pΩ^{∗}).  (20)

Therefore, the problem is reduced to estimating N(pΩ^{∗}).

Let T ≥ 1. We further restrict the value of T in (39) and (49), and finally specify it in (53). Now we explain how T is used. We represent ℝ^{mr} as a disjoint union of cubes:

ℝ^{mr} = ⋃_{k_{1},... ,k_{mr}∈ℤ}(k_{1} \frac{p}{T} + [0, \frac{p}{T} )) × … × (k_{mr} \frac{p}{T} + [0, \frac{p}{T} )) .  (21)

Let I_{p/T} (pΩ^{∗}) be the union of cubes in (21) contained in pΩ^{∗}, and E_{p/T} (pΩ^{∗}) the union of cubes in (21) intersecting pΩ^{∗}. Then

I_{p/T} (pΩ^{∗}) ⊆ pΩ^{∗} ⊆ E_{p/T} (pΩ^{∗}) .  (22)

In below, we will show that each of the two sets I_{p/T} (pΩ^{∗}) and E_{p/T} (pΩ^{∗}) contain

T^{mr}V(Ω^{∗}) + O_{Ω}(T^{mr−1})  (23)

cubes of the tessellation (21). Then in (48) we will estimate the number of integral points each such a cube contains. This in turn will yield (51), an estimate for N(pΩ^{∗}).

2.3 Proof of (23)

For the purpose of proving (23), we may assume that the cubes in (21) are replaced with the closed cubes. Indeed, since pΩ^{∗} is closed, the number A_{T} of cubes in (21) contained in pΩ^{∗} equals the number A_{c,T} of the closures of cubes in (21) contained in pΩ^{∗}. Also, the number B_{T} of cubes in (21) intersecting pΩ^{∗} is at most the number B_{c,T} of the closures of cubes in (21) intersecting pΩ^{∗}. Then A_{c,T} = A_{T} ≤ B_{T} ≤ B_{c,T}. Hence, if we prove that both A_{c,T} and B_{c,T} are T^{mr}V(Ω^{∗}) + O_{Ω}(T^{mr−1}), then the same estimate will hold for A_{T} and B_{T}.

Now we rescale pΩ^{∗} to Ω^{∗} ⊆ [−1, 1]^{mr}, the cube [−1, 1]^{mr} being tessellated with the cubes of side length 1/T. We will refer to them in this section only as the “tessellation cubes”. From Ω ∈ D_{mr−1} we conclude that Ω^{∗} ∈ D_{mr}. Then Ω^{∗} ∈ D_{mr}(h) for some positive integer h. We will prove the estimate (23) in three steps below. We will also assume that T is chosen large enough to satisfy the conditions of Definition 3. This is legitimate, because from (53) it will follow that T → +∞ as p → +∞.

1. Let I_{T} (Ω^{∗}) be the union of the tessellation cubes contained in Ω^{∗}. Then

(a) I_{T} (Ω^{∗}) ⊆ Ω^{∗}.

(b) I_{T} (Ω^{∗}) ∈ D_{mr}(h) by Condition 5c of Definition 3.

INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 5 (2005), #A04 8

(c) N_{T} (I_{T} (Ω^{∗})) = N_{T} (Ω^{∗}) + O_{Ω} (T^{mr−1}), where N_{T} (·) denotes the number of points whose coordinates are multiples of 1/T. This claim follows from Condition 5a of Definition 3.

(d) The number of cubes of (21) contained in I_{p/T} (pΩ^{∗}) is V_{T} (I_{T} (Ω^{∗})) = T^{mr}V (I_{T} (Ω^{∗})) = A_{T}, where V_{T} (·) denotes the (mr-dimensional) volume of the set measured with respect to the unit length 1/T.

(e) By the inequality (7), rescaled with respect to the unit length 1/T,

|V_{T} (I_{T} (Ω^{∗})) − N_{T} (Ω^{∗})| ≤ O_{Ω} (T^{mr−1}) + ∑_{j=0}^{mr−1} V_{T,j} (I_{T} (Ω^{∗})) h^{mr−1} ≤

O_{Ω} (T^{mr−1}) + ∑_{j=0}^{mr−1} T^{j}V_{j} (Ω^{∗}) h^{mr−j} = O_{Ω} (T^{mr−1}) . (24)

In the next step, we obtain a similar estimate for |V_{T} (E_{T} (Ω^{∗})) − N_{T} (Ω^{∗})|, where E_{T} (Ω^{∗}) is the union of the tessellation cubes intersecting Ω^{∗}.

2. We observe the following properties of E_{T} (Ω^{∗}).

(a) Ω^{∗} ⊆ E_{T} (Ω^{∗}).

(b) E_{T} (Ω^{∗}) ∈ D_{mr}(h) by Condition 5c of Definition 3.

(c) N_{T} (E_{T,ε} (Ω^{∗})) = N_{T} (Ω^{∗}), by Condition 5b of Definition 3.

(d) V_{T} (E_{T,ε} (Ω^{∗})) = T^{mr}V (E_{T,ε} (Ω^{∗})) → T^{mr}V (E_{T} (Ω^{∗})) = B_{T} as ε → 0. Also B_{T} is equal to the number of cubes of (21) contained in E_{T} (Ω^{∗})

(e) From E_{T} (Ω^{∗}) ⊆ [−1, 1]^{mr}, we conclude that there is a positive number κ (independent of T) such that V_{j} (E_{T} (Ω^{∗})) ≤ κ for all j.

(f) By the inequality (7),

|V_{T} (E_{T,ε} (Ω^{∗})) − N_{T} (Ω^{∗})| ≤ κ ∑_{j=0}^{mr−1} T^{j}h^{mr−j}.  (25)

Therefore, taking the limit as ε → 0, we obtain

|V_{T} (E_{T} (Ω^{∗})) − N_{T} (Ω^{∗})| = O_{Ω} (T^{mr−1}) .  (26)

3. By the triangle inequality, (24), and (26), we conclude that

|A_{T} − B_{T} | = O_{Ω} (T^{mr−1}) .  (27)

At this point, we would like to recall that by Condition 4 of Definition 3, V (I_{T} (Ω^{∗})) − V (Ω^{∗}) = O_{Ω} (1/T) and V (E_{T} (Ω^{∗})) − V (Ω^{∗}) = O_{Ω} (1/T). This remark together with (27) finally proves (23).

Next section will deal with the number 𝔑(J) of integral points x of N_{C} such that px^{∗} belongs to a cube of the subdivision (21). The estimation of μ_{C}(Ω) will be done in 2.4.3.

2.4 Estimating 𝔑(J)

Let J be a cube in the subdivision (21). Then

𝔑(J) = #{x ∈ N_{C} | px^{∗} = (x_{1}, x_{2} − x_{1}, …, x_{mr} − x_{mr−1}) ∈ J}.  (28)

We write the cube J as a direct product of intervals:

J = T_{1} × … × T_{mr}.  (29)

This allows us to express 𝔑(J) in terms of the characteristic functions χ_{T_{j}} of the intervals T_{1}, …, T_{mr} as follows:

𝔑(J) = ∑_{y=(y_{1},... ,y_{mr})∈J} χ_{T_{1}}(y_{1})χ_{T_{2}}(y_{2}) … χ_{Tmr} (y_{mr}) =

{recall that y_{1} = x_{1}, y_{2} = x_{2} − x_{1}, …, y_{mr} = x_{mr} − x_{mr−1}}

∑_{x=(x_{1},... ,x_{mr})∈N_{C}} χ_{T_{1}}(x_{1})χ_{T_{2}}(x_{2} − x_{1}) … χ_{Tmr} (x_{mr} − x_{mr−1}). (30)

Now we can write the characteristic function χ_{T} (x) as an exponential sum. In the the next formula, we assume that x ∈ J, where J is an interval (closed or not) of length ≤ p.

χ_{T} (x) = p^{−1} ∑_{z∈T} ∑_{k(modp)} e(k(x − z)/p),  (31)

where the sum ∑_{z∈T} is taken over the integral points of T . We substitute (31) in (30), and change the order of summation:

𝔑(J) = p^{−mr} ∑_{k=(k_{1},... ,k_{mr})∈𝔽_{p}^{mr}} ∏_{j=1}^{mr} ( ∑_{y_{j}∈T_{j}} e(−k_{j}y_{j}/p)) S_{k},  (32)

where

S_{k} = ∑_{x∈N_{C}} e(L_{k}(x)/p)  (33)

and, in turn,

L_{k}(x) = k_{mr}x_{mr} + ∑_{s=1}^{mr−1}(k_{s} − k_{s+1})x_{s}.  (34)

Since we require that C does not lie in a hyperplane of 𝔸^{mr}(𝔽̄_{p}), the linear form L_{k}(x) is constant on C if and only if k_{1} = … = k_{mr} = 0. (At this point we would like to recall that that our hyperplanes are assumed to be affine). We next show that the sum of the terms in (32) with k_{1} = … = k_{mr} = 0 is the main term in (32). The sum of the remaining terms will be proved to be of the lower order of magnitude, and therefore is the error term. We denote the main term and the error term by M(J) and E(J), respectively. Then

𝔑(J) = M(J) + E(J)  (35)

We will make this formula more precise below, with the final result stated in (50) of 2.4.3.

2.4.1 The main term in (35)

Each T_{j} has length p/T (we denote this by |T_{j}| = p/T); therefore

V(J) = (p/T)^{mr}.  (36)

By (32) and (36),

\frac{M(J)}{p^{−mr}#C(𝔽_{q})} = ∏_{j=1}^{mr} N(T_{j}) = ∏_{j=1}^{mr} (p/T + O_{m,r,d}(1)) = V(J) (1 + O_{m,r,d} (T/p)) . (37)

Hence, substituting (36) in (37), we have

M(J) = T^{−mr}#C(𝔽_{q}) (1 + O_{m,r,d} (T/p)) .  (38)

At this point we further assume that

1 ≤ T ≤ p^{1/2}.  (39)

We substitute (39) in (38):

M(J) = T^{−mr}#C(𝔽_{q}) (1 + O_{m,r,d} (p^{−1/2})) .  (40)

2.4.2 The error term in (35)

We need to estimate the error term

E(J) = p^{−mr} ∑_{(0,... ,0)≠(k_{1},... ,k_{mr})∈𝔽_{p}^{mr}} ∏_{j=1}^{mr} ( ∑_{y_{j}∈T_{j}} e (\frac{−k_{j}y_{j}}{p} )) S_{k}.  (41)

To estimate the sums of the form ∑_{y_{j}∈T_{j}} in (41), we need the following well known lemma.

Lemma 3 Let ∥·∥ denote the distance to the nearest integer on the real line. Then for any real a and integers l ≥ 1 and n, we have

| ∑_{j=n+1}^{n+l} e(ja)| ≤ min (l, \frac{1}{2∥a∥})  (42)

Remark 6 We assume that if a = 0, then min (l, 1/(2∥a∥)) = l.

Now we return to the question of estimating the sums of the form ∑_{y_{j}∈T_{j}} in (41). By Lemma 3,

| ∑_{y_{j}∈T_{j}} e (−k_{j}y_{j}/p)| ≤ min \{|T_{j}| + 1, \frac{1}{2∥k_{j}/p)∥}\} .  (43)

We use the inequalities 1/x ≤ 2/(x + 1) and 1 + 1/x ≤ 2, for x ≥ 1, and substitute (43) in (41):

E(J) ≤ ∑_{(0,... ,0)≠(k_{1},... ,k_{mr})∈𝔽_{p}^{mr}} ∏_{j=1}^{mr} \frac{1}{p} (min \{p + 1, \frac{p}{|k_{j}|}\}) |S_{k}| ≪_{mr}

∑_{(0,... ,0)≠(k_{1},... ,k_{mr})∈𝔽_{p}^{mr}} ∏_{j=1}^{mr} (\frac{1}{1 + |k_{j}|}) |S_{k}| . (44)

Next, for each x ≠ 0, our hypotheses on the curve C (see also Remark 5 above) allow us to apply Bombieri’s inequality (see [1], Th. 6, p. 97):

S_{k} = O_{m,r,d} (q^{1/2}) .  (45)

We substitute (45) in (44):

|E(J)| = O_{m,r,d} (q^{1/2} ln^{mr} q) .  (46)

2.4.3 Conclusion

We recall that Weil’s theorem states that (see also Remark 5 above)

#C(𝔽_{q}) = #N_{C} = q + O_{m,r,d} (q^{1/2}) .  (47)

We substitute (40), (46), and (47) in (35):

𝔑(J) = T^{−mr} (1 + O_{m,r} (p^{−1/2})) (q + O_{m,r,d} (q^{1/2})) + O_{m,r,d} (q^{1/2} ln^{mr} q) =

qT^{−mr} (1 + O_{m,r,d} (q^{\frac{1}{2}} ) + O_{m,r,d} (q^{1− \frac{1}{2m}} ) + O_{m,r,d} (q^{\frac{1}{2} − \frac{1}{2m}} )) + O_{m,r,d} (q^{\frac{1}{2}} ln^{mr} q) . (48)

At this point, we further restrict T as follows:

\frac{q^{1− \frac{1}{2m}}}{T^{mr}} ≤ q^{\frac{1}{2}} ln^{mr} q.  (49)

This allows us to simplify (48) in the following way:

𝔑(J) = q/T^{mr} + O_{m,r,d} (q^{1/2} ln^{mr} q) .  (50)

2.5 Estimating μ_{C}(Ω)

The formula (20) allows us to estimate the numerator in (5) as the product of the right-hand sides of (23) and (50):

#{x ∈ N_{C} | x̃ ∈ Ω} = N(pΩ^{∗}) =

(q/T^{mr} + O_{m,r,d} (q^{1/2} ln^{mr} q)) (T^{mr}V(Ω^{∗}) + O_{Ω} (T^{mr−1})) =

qV(Ω^{∗}) + O_{m,r,d,Ω} (q/T) + O_{m,r,d,Ω} (T^{mr}q^{1/2} ln^{mr} q) . (51)

Next, the denominator in (5) can be estimated by (47). This yields

μ_{C}(Ω) = V(Ω^{∗}) + O_{m,r,d,Ω} (T^{mr}q^{−1/2} ln^{mr} q) + O_{m,r,d,Ω}(1/T).  (52)

Now we specify T as the root of the equation T^{mr}q^{−1/2} ln^{mr} q = 1/T:

T = q^{\frac{1}{2(mr+1)}} ln^{\frac{−mr}{mr+1}} q.  (53)

(We remark that (39), (49), and (53) agree with each other.) The value of T given by (53) allows us to have only one error term in (52):

μ_{C}(Ω) = V(Ω^{∗}) + O_{m,r,d,Ω} (q^{\frac{−1}{2(mr+1)}} ln^{\frac{mr}{mr+1}} q) ,  (54)

and we may apply (18). This finally proves Theorem 1.

References

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