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\begin{document}
\vspace*{-60pt} 
\centerline{\smalltt INTEGERS: 
 \smallrm ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY \smalltt 5 
(2005), \#A04} 
\vskip 50pt


\begin{center}
 \uppercase{\bf On the distribution of distances between the points of affine curves
 over finite fields}
 \vskip 20pt
 {\bf B.V.~Petrenko}\\
 {\smallit Department of Mathematics, Texas A\&M University, College Station, Texas 77843-3368,
USA}\\
 {\tt petrenko@math.tamu.edu}
 \end{center}
 \vskip 30pt
 \centerline{\smallit Received: 9/23/04, Accepted: 2/5/05, Published: 2/15/05}
 \vskip 30pt

 \centerline{\bf Abstract}
\noindent
Let $\mathbb{F}_q$ be a finite field with $q$ elements, 
$\bar{\mathbb{F}}_q$ an algebraic closure of $\mathbb{F}_q$,
and 
$\mathbb{A}^n\left(\bar{\mathbb{F}}_q \right)$ an $n$-dimensional affine
space over $\bar{\mathbb{F}}_q$.
Let $\mathcal{C}$ be an affine absolutely irreducible curve in 
$\mathbb{A}^n\left(\bar{\mathbb{F}}_q \right)$.
We interpret the points of $\mathcal{C}$ over $\mathbb{F}_q$
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 $\mathcal{C}$ in $[-1,1]^{n-1}$ provided 
the characteristic $p$ of $\mathbb{F}_q$ is large,
while $n$, $\log_p q$ are fixed, and the degree of $\mathcal{C}$ is bounded.
When $p = q$, this becomes a recent result of Cobeli and Zaharescu.


\footnotesize
\noindent
{\it Keywords}: finite field, curves over finite fields, distribution of points,
Bombieri's inequality, principle of Lipschitz,  Weil's theorem.


\noindent
{\it AMS~Subject~Classification}: 11G20, 11L05, 11L07, 11T23.

\normalsize


\pagestyle{myheadings}
 \markright{\smalltt INTEGERS: \smallrm ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY \smalltt
5 (2005), \#A04\hfill}

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 \baselineskip=15pt 
 \vskip 30pt


\section*{\normalsize 1. Introduction}
\addtocounter{section}{+1}

This paper gives a generalization the main result of Cobeli and Zaharescu \cite{Cobeli}.
The main result of \cite{Cobeli} generalizes that of Zheng \cite{Zheng} and partially
generalizes that of Zhang \cite{Zhang}. This will be explained in detail below.
 
Let $p$ be prime number, and $q = p^m$. 
Let $\mathcal{C}$ be a curve of degree $d$ in an affine space $\mathbb{A}^r(\bar{\mathbb{F}}_q)$, 
where $\bar{\mathbb{F}}_q$ is an algebraic closure of  a finite field with
$q$ elements $\mathbb{F}_q$. 
The goal of this paper is to study the distribution of the points of
$\mathcal{C}$ in the cube $[-1, 1]^{mr-1} \subseteq \mathbb{R}^{mr-1}$.

 
Let us begin by explaining how the points of $\mathcal{C}$ are interpreted as points
of $[-1,1]^{mr-1}$. The resulting set will be called 
$\widetilde{\mathcal{N}_{\mathcal{C}}}$. We assume that
$\mathcal{C}$ is not contained in a hyperplane
\footnote{In this paper, all hyperplanes are assumed to be affine, 
i.e.~given by an equation $\sum_{k=1}^{mr} \alpha_k x_k = c$ for some 
$c, \alpha_1, \ldots, \alpha_{mr} \in \mathbb{F}_p$.}
of $\mathbb{A}^{mr}(\bar{\mathbb{F}}_p)$. 
There is
a bijection between $\mathbb{F}_p =  \mathbb{Z} / p \mathbb{Z}$ and 
$\{ 0, \ldots, p-1 \}$ via $l + p \mathbb{Z} \mapsto l$. Then $\mathbb{F}_q$ can be identified
with $\{ 0, \ldots, p-1 \}^m$, and it makes sense to consider the set  
\begin{equation}
\mathcal{N}_{\mathcal{C}} = 
\{\mathbf{x} = (x_1, \ldots, x_{mr}) \in \{0, \ldots, p-1\}^{mr} 
\mid \mathbf{x} \in \mathcal{C}\}.
\end{equation}
In general, each such an identification of $\mathcal{C}$
with $\mathcal{N}_{ \mathcal{C}}$ corresponds to a basis
of $\mathbb{F}_q$ over $\mathbb{F}_p$. Nevertheless, the main result of this paper,
Theorem \ref{Great}, is independent of this choice. 


Sometimes we will prefer to think of $\mathcal{N}_{ \mathcal{C}}$
as a subset of $\mathbb{F}_p^{mr}$. This is legitimate because 
$\mathcal{N}_{ \mathcal{C}}$ will be regarded as the domain of some $p$-periodic functions
related to the exponential function $e^{\frac{2 \pi i x}{p}}$.


\begin{definition} We will write $e( t)$ instead of
$e^{2 \pi i t}$.
\end{definition}
Next we consider the map
\begin{equation}
\widetilde\,: \mathbb{R}^{mr} \to \mathbb{R}^{mr-1},
\, \mathbf{x} = (x_1, \ldots, x_{mr}) \mapsto
\mathbf{\widetilde{x}} = (\widetilde{x_1}, \ldots, \widetilde{x_{mr-1}}),
\end{equation}
where
\begin{equation}
\widetilde{x_j} = \frac{x_{j+1} - x_j}{p}.
\end{equation}
In this paper we obtain some asymptotic results about the 
the set
\begin{equation}
\widetilde{\mathcal{N}_{ \mathcal{C}}}
\end{equation} 
endowed with the probability measure $\mu_{\mathcal{C}}$
defined by the formula
\begin{equation}\label{Prob}
\mu_{\mathcal{C}}(\Omega) = 
\frac{\#\{\textbf{x} \in \mathcal{N}_{\mathcal{C}}\mid \widetilde{\textbf{x}} \in \Omega\}}
{\# \mathcal{N}_{ \mathcal{C}}}.
\end{equation}

By placing some restrictions on $\mathcal{C}$, 
the denominator of (\ref{Prob}) may be estimated by a well known theorem of A.~Weil. Therefore, the
point of this paper is to estimate the numerator of (\ref{Prob}). This will involve
the theorems of Bombieri (\cite{Bombieri}, Th.~6, p.~97), Davenport \cite{Davenport}, and Weil.
Our proof extends that of Cobeli and Zaharescu \cite{Cobeli}.

\smallskip

Before formulating the main result of this paper, Theorem \ref{Great}, we would like to describe 
a certain class $\mathcal{D}_n(h)$ of subsets in $\mathbb{R}^n$.
(A more general class has been introduced in an important paper
of Davenport \cite{Davenport}.) 
Our Definition \ref{Defini} below is more restrictive than
Davenport's because  we additionally impose Conditions \ref{it5} and \ref{it6}. 
 
\begin{remark}
It would be desirable not only to relax these restrictions in Definition \ref{Defini}, but
also to generalize the main result of \cite{Davenport}.
\end{remark} 

To make the formulation of Definition \ref{Defini} less  cumbersome, we
introduce the following auxiliary

\begin{definition}\label{PreDefini} Let $X \subseteq \mathbb{R}^n$ and
$\tau > 0$. We introduce the following tessellation of $\mathbb{R}^n$:
\begin{equation}\label{Tess1}
\mathbb{R}^n = \bigcup_{\textbf{t} \in \mathbb{Z}^n} (1/ \tau)\textbf{t} + [0, 1/ \tau]^n.
\end{equation}

\begin{enumerate}
\item We define the sets $\mathcal{I}_{\tau}(X)$ and $\mathcal{E}_{\tau}(X)$ to be the unions of cubes in (\ref{Tess1})
contained in $X$ and intersecting $X$, respectively.



\item For any $\varepsilon>0$, we define the set
$\mathcal{E}_{\tau, \varepsilon}\left( X \right) = X \cup \Delta_{\tau,\varepsilon}$ , where
$\Delta_{\tau, \varepsilon}$ consists of the points of $\mathcal{E}_{\tau}\left( X \right)$ whose 
standard distance
to the boundary of $\mathcal{E}_{\tau}\left( X \right)$ is at least $\varepsilon$.
\end{enumerate}
\end{definition}

Now we are ready to give the following

\begin{definition}\label{Defini} Let $X \subseteq \mathbb{R}^n$ and $h$ a positive integer. Then
$X \in \mathcal{D}_n(h)$ if all of the following conditions are satisfied:
\begin{enumerate}
\item\label{it1} $X$ is compact.

\item\label{it2} Any line parallel to one of the $n$ coordinate axes intersects
$X$ in at most $h$ intervals.

\item\label{it3} The same is true for any $m \in \{1, \ldots, n-1 \}$ and any projection
of $X$ on any of the $m$-dimensional coordinate subspaces.

\item\label{it5}
Let $V(\cdot)$
denote the ($n$-dimensional) volume of the set. Then
$V(X)$ 
exists and has the following properties: 
$V(X) - V \left( \mathcal{I}_{\tau}(X)\right) = O_X(1/\tau)$ and
$V(X) - V \left( \mathcal{E}_{\tau}(X)\right) = O_X(1/\tau)$ as $\tau \to + \infty$.

\item\label{it6} 
There exists $\tau_0 \geq 1$, depending on $X$, such that for any $\tau \geq \tau_0$, 
\begin{enumerate}
\item\label{Interior} All but $O_X(\tau^{n-1})$ vertices
of the grid (\ref{Tess1}) in $\mathcal{I}_{\tau}(X)$ possess the following property.  
For a vertex $v$ there exists a cube $C_v$ of (\ref{Tess1})
with $v \in C_v \subseteq X$.

\item\label{Exterior}  
Any vertex $w \in \mathcal{E}_{\tau}(X)\diagdown X$ of the grid (\ref{Tess1}) 
is a boundary point of $\mathcal{E}_{\tau}(X) $.

\item\label{Peace} The sets 
$\mathcal{I}_{\tau}(X)$, $\mathcal{E}_{\tau}(X)$, and 
$\mathcal{E}_{\tau, \varepsilon}\left( X \right)$, for any $\varepsilon > 0$,
satisfy Conditions \ref{it2} and \ref{it3} of this definition.

\end{enumerate}
\end{enumerate}
\end{definition}


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

\begin{definition}
$\mathcal{D}_n = \bigcup_{h=1}^{\infty}\mathcal{D}_n(h)$.
\end{definition}


\begin{definition}
$\mathcal{D} = \bigcup_{n=1}^{\infty}\mathcal{D}_n$.
\end{definition}

\smallskip

At this point, we would like to recall the main result of Davenport \cite{Davenport}.
We state it for a possibly smaller class of sets, as explained above.
\begin{remark}\label{Daven}
Let $X \in \mathcal{D}_n(h)$, $N(X)$ 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
\begin{equation} \label{Daven1}
\left| N(X) - V(X) \right| \leq
\sum_{j=0}^{n-1} h^{n-j} V_j(X).
\end{equation} 
\end{remark}

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 $\mu_{ \mathcal{C}}$
in Theorem \ref{Great}.

\begin{definition}
\begin{multline}
g_{n}(t_1, \ldots, t_{n}) = 
\max \left\{ 
0, \min_{1\leq k \leq n} \left\{ 1, 1- \sum_{s=1}^k t_s \right\}+
\min_{1\leq k \leq n} \left\{ 0, \sum_{s=1}^k t_s \right\} \right\}.
\end{multline}
\end{definition} 

The main result of this paper is the following

\begin{theorem}\label{Great} Let
$\{q_j\}$ an increasing sequence of powers of primes 
$\{p_j\}$ with $\log_{p_j}q_j   =  m  = \text{const}$.
Let $\mathcal{C}_j$ be an irreducible affine algebraic curve in
$\mathbb{A}^{r}(\bar{\mathbb{F}}_{q_j})$  
of degree $\leq d = \text{const}$. Suppose that $\mathcal{C}_j$ is not contained in a hyperplane
of $\mathbb{A}^{mr}(\bar{\mathbb{F}}_{p_j})$.
Then for any  $\Omega \in \mathcal{D}_{mr-1}$,
\begin{equation}\label{MainTh}
\mu_{ \mathcal{C}_j}(\Omega) = 
\int_{\Omega}  
g_{mr-1}(\textbf{t})d \textbf{t} +
O_{m,r,d, \Omega} \left(q_j^{-\frac{1}{2(mr+1)}} \ln^{\frac{mr}{mr+1}}q_j \right)
\end{equation}
as $j \to +\infty$. (Here $\ln(\cdot)$ denotes the logarithm to base $e$.)
\end{theorem}
 
In other words, the measures $\mu_{ \mathcal{C}_j}$ weakly converge,
as $j \to +\infty$, to $\mu_{mr-1}$ with density
function $g_{mr-1}$. We believe
that Theorem \ref{Great} is of interest because $g_{mr-1}$ is independent
of $d$, $\{q_j\}$, $\{\mathcal{C}_j\}$.

\begin{corollary}\label{SalesPitch}
If $m=1$, then Theorem \ref{Great} becomes the main result of Cobeli and Zaharescu \cite{Cobeli}.
In turn, let $f \in \mathbb{Z}[x,y]$ be of degree $d\geq 2$, and suppose that $f$ is absolutely irreducible 
modulo all large primes. 
If $\mathcal{C}_j$ is a plane curve obtained by reducing 
$f $ modulo $p_j$, then we obtain the main result of Zheng \cite{Zheng}.
Finally, if the curve is of the form $f(x,y) = xy-1$, then we partially recover the main
result of Zhang \cite{Zhang}.
\end{corollary}


\begin{remark} Theorem \ref{Great} may be strengthened as follows:
$m$ and $r$  may be allowed
to depend on the curve, provided they are bounded. 
This formulation of Theorem \ref{Great} leaves only finitely
many possibilities for the values of $m$ and $r$, and therefore the same proof
is valid in this situation as well.
\end{remark}

\begin{remark}\label{CondProb}
It may be of interest to prove that the formula 
\[\nu_{\mathcal{C}}(\Omega) = 
\frac{\#  \widetilde{\mathcal{N}_{\mathcal{C}} } \cap \Omega }
{\#\widetilde{\mathcal{N}_{\mathcal{C}} }}\]
defines an asymptotically well defined conditional probability measure on $\mathbb{R}^{mr-1}$, and
to compare its asymptotic behavior to that of $\mu_{\mathcal{C}}$.
\end{remark}

\begin{remark}\label{Lou} By Weil's theorem, the number of points in the projective closure
of the curve $\mathcal{C}_j$ is $q_j + O_{m,r,d}\left(q_j ^{1/2} \right)$. By the assumptions
in Theorem \ref{Great}, however, the same estimate holds for the curve $\mathcal{C}_j$ itself. 
A similar 
observation about the more general Lang-Weil estimates of \cite{Lang} has been made 
in \cite{Chat},~p.~120. This observation also allows us to apply the result of 
Bombieri (\cite{Bombieri}, Th.~6, p.~97).
\end{remark}

\noindent
{\textbf{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 \cite{Chat}. Jeremy Tyson has very helpfully commented
on the definition of $\mathcal{D}_n(h)$.} 


\section*{\normalsize 2. Proof of Theorem \ref{Great}}
\addtocounter{section}{+1}
{\em To simplify the notation, we will denote 
$p_j$, $q_j$, $\mathcal{C}_j$
by $p$, $q$, $\mathcal{C}$, respectively.} 

\subsection*{\normalsize 2.1 The map $^*$.}
\addtocounter{subsection}{+1}
If $\left(x_1, \ldots, x_{mr} \right) \in \mathbb{R}^{mr}$, then define 
\begin{equation}
y = \frac{x_1}{p},\, t_1 = \frac{x_2 - x_1}{p},\, \ldots,\, 
t_{mr-1} = \frac{x_{mr} - x_{mr-1}}{p}.
\end{equation}
Let
$(x_1, \ldots, x_{mr}) \in \mathcal{N}_{ \mathcal{C}}$, then 
$0 \leq x_1, \ldots, x_{mr} \leq p-1$.
Therefore
\begin{equation}\label{Useful1}
0 \leq y, \,
y+ \sum_{j=1}^{k}t_j = \frac{x_{k+1}}{p}  \leq 1, ~\text{where} ~ k \in \{1, \ldots, mr-1\}.
\end{equation}
This can be restated as follows:
\begin{equation}\label{Useful}
0 \leq y \leq 1, ~  - \sum_{j=1}^{k}t_j \leq y \leq 1 - \sum_{j=1}^{k}t_j, ~
k \in \{1, \ldots, mr-1\}. 
\end{equation}
Based on these considerations, for $\Omega \in \mathcal{D}_{mr-1}$,
{\em we define the set $\Omega^*$} by
\begin{equation}\label{Hyper}
\Omega^* = 
\{ (y, t_1, \ldots, t_{mr-1}) \in \mathbb{R} \times \Omega
\mid y, t_1, \ldots, t_{mr-1} \,\,
\text{satisfy} \,\, (\ref{Useful})  \}.
\end{equation}
We remark that $\Omega^* \in \mathcal{D}_{mr}$.

From (\ref{Hyper}), we see that the set $\Omega^*$
can be described as a cylinder bounded by some two hypersurfaces,
$h(t_1, \ldots, t_{mr-1})$ and $H(t_1, \ldots, t_{mr-1})$,
as follows:
\begin{equation}\label{Hyper1}
\Omega^* = 
\left\{
(y, \textbf{t}) \in \mathbb{R}^{mr} \mid 0 \leq y \leq 1, \,
\textbf{t} \in \Omega,\, 
h(\textbf{t}) \leq H(\textbf{t})
\right\}.
\end{equation}
We proceed to find equations of these hypersurfaces based on (\ref{Useful}). 

1.  For a given $\textbf{t} \in \Omega$,
the smallest value of $y$ is
\begin{multline}\label{Hyper2}
h(t_1, \ldots, t_{mr-1})  = \max \{ 0, -t_1, \ldots, -t_1, + \ldots + t_{mr-1}\}= \\ 
 - \min \{0, t_1, \ldots, t_1+ \ldots + t_{mr-1} \}.
\end{multline}

 
2. For a given $\textbf{t} \in \Omega$,
the largest value of $y$ is
\begin{equation}
H(t_1, \ldots, t_{mr-1}) = \min\{1, 1 -t_1, \ldots, 1  -t_1- \ldots - t_{mr-1} \}.
\end{equation}
We see that
\begin{equation}
 g_{mr-1}(\textbf{t}) = 
\max\left\{0,~H(\textbf{t}) - h(\textbf{t}) \right\}.
\end{equation}
Therefore, the volume of $\Omega^*$ may be written as follows:
\begin{equation}\label{Vol1}
\text{V}(\Omega^*) = \int_{\Omega}  
g_{mr-1}(t_1, \ldots, t_{mr-1})d t_1 \ldots dt_{mr-1}.
\end{equation}
We will use this formula in (\ref{FinMeas}) of \ref{General}.

\subsection*{\normalsize 2.2 Main idea}
\addtocounter{subsection}{+1}
We will use several rescalings below. One could think of them as changing the unit of length.
We firstly pass from $\Omega^*$ to $p\Omega^*$, because
there is a bijection between $\mathcal{N}_{ \mathcal{C}}$
and the set of integral points
of $p\Omega^*$, namely, 
\begin{equation}\label{Crucial}
\left(x_1, \ldots, x_{mr} \right) = 
\textbf{x} \mapsto p \textbf{x}^* = \left(x_1, x_2 - x_1, \ldots, x_{mr} - x_{mr-1} \right).
\end{equation}
In particular, we may express the numerator of (\ref{Prob}) as follows:
\begin{equation}\label{Connection}
 \#\{\textbf{x} \in \mathcal{N}_{\mathcal{C}}\mid \widetilde{\textbf{x}} \in \Omega\} = 
  N\left( \text{preimage of $\Omega$ under \,$\widetilde{}$}\,\, \right) = N(p \Omega^*).
\end{equation}
Therefore, the problem is reduced to estimating $N(p \Omega^*)$. 

\smallskip

Let $T \geq 1$. 
We further restrict the value of $T$ in (\ref{Assum1}) and (\ref{Serf}), and
finally specify it in (\ref{FinT}). Now we explain how $T$ is used. 
We represent $\mathbb{R}^{mr}$ as a disjoint
union of cubes:
\begin{equation}\label{Tess} 
\mathbb{R}^{mr} = \bigcup_{k_1, \ldots, k_{mr} \in \mathbb{Z}}
\left(k_1 \frac{p}{T} + \left[0, \frac{p}{T}\right) \right)\times 
\ldots \times \left(k_{mr} \frac{p}{T} + \left[0, \frac{p}{T}\right) \right).
\end{equation}

 
Let $\mathcal{I}_{p/T}\left(p \Omega^* \right)$ 
be the union of cubes in (\ref{Tess}) contained in $p\Omega^*$,
and $\mathcal{E}_{p/T}\left(p \Omega^* \right)$ the union of cubes in (\ref{Tess})
intersecting $p\Omega^*$. Then 
\begin{equation}\label{Sandwich}
\mathcal{I}_{p/T}\left(p \Omega^* \right) \subseteq p\Omega^* \subseteq 
\mathcal{E}_{p/T}\left(p \Omega^* \right).
\end{equation}
In \ref{Stripping} below, we will show that each of the two sets $\mathcal{I}_{p/T}\left(p \Omega^* \right)$
and $\mathcal{E}_{p/T}\left(p \Omega^* \right)$ contain 
\begin{equation}\label{Sandwich1}
T^{mr} \text{V}({\Omega^*}) + O_{\Omega}(T^{mr-1})
\end{equation}
cubes of the tessellation (\ref{Tess}). Then in (\ref{PreCubfinerror}) we will estimate 
the number of integral points each such a cube contains. This in turn will yield (\ref{Num1}),
an estimate for $N(p \Omega^*)$.


\subsection*{\normalsize 2.3 Proof of (\ref{Sandwich1})}\label{Stripping} 
\addtocounter{subsection}{+1}
{\em For the purpose of proving (\ref{Sandwich1}), we may assume that the cubes
in (\ref{Tess}) are replaced with the closed cubes}. Indeed, since
$p \Omega^*$ is closed, the number $\mathcal{A}_T$ of cubes in (\ref{Tess})
contained in $p \Omega^*$ equals the number $\mathcal{A}_{c,T}$ of the closures
of cubes in (\ref{Tess}) contained in $p \Omega^*$. Also,
the number $\mathcal{B}_T$ of cubes in (\ref{Tess}) intersecting
$p \Omega^*$ is at most the number $\mathcal{B}_{c,T}$ of the closures
of cubes in (\ref{Tess}) intersecting $p \Omega^*$.
Then 
$\mathcal{A}_{c,T} = \mathcal{A}_T \leq \mathcal{B}_T \leq \mathcal{B}_{c,T}$.
Hence, if we prove that both $\mathcal{A}_{c,T}$ and $\mathcal{B}_{c,T}$
are $T^{mr} \text{V}({\Omega^*}) + O_{\Omega}(T^{mr-1})$, then 
the same estimate will hold for $\mathcal{A}_T$ and $\mathcal{B}_T$.

\smallskip

Now we rescale $p \Omega^*$ to $\Omega^* \subseteq [-1,~1]^{mr}$,
the cube $[-1,~1]^{mr}$ being tessellated with the cubes
of side length $1/T$. {\em We will refer to them in this section only as 
the ``tessellation cubes".}  From $\Omega \in \mathcal{D}_{mr-1}$ we conclude
that $\Omega^* \in \mathcal{D}_{mr}$. Then $\Omega^* \in \mathcal{D}_{mr}(h)$
for some positive integer $h$. We will prove the estimate (\ref{Sandwich1})
in three steps below. We will also assume that $T$ is chosen large enough
to satisfy the conditions of Definition \ref{Defini}. This is legitimate,
because from (\ref{FinT}) it will follow that $T \to +\infty$ as $p \to +\infty$. 


\textbf{1.} Let $\mathcal{I}_{T}\left( \Omega^* \right)$ be the union of the tessellation cubes contained 
in $\Omega^*$. Then 

\smallskip

(a) $\mathcal{I}_{T}\left( \Omega^* \right) \subseteq \Omega^*$.

\smallskip

(b) $\mathcal{I}_{T}\left( \Omega^* \right) \in \mathcal{D}_{mr}(h)$ by Condition \ref{Peace}
of Definition \ref{Defini}.

\smallskip

(c) $N_T\left( \mathcal{I}_{T}\left( \Omega^* \right) \right) = 
N_T\left( \Omega^* \right) + O_{\Omega} \left(T^{mr-1} \right)$, 
where
$N_T(\cdot)$ denotes the number of points whose coordinates are multiples
of $1/T$. This claim follows from Condition \ref{Interior} of Definition \ref{Defini}.

\smallskip

(d)  {\em The number of cubes of (\ref{Tess}) contained
in $\mathcal{I}_{p/T}\left(p \Omega^* \right)$} is
$V_T\left( \mathcal{I}_{T}\left( \Omega^* \right) \right) = 
T^{mr} V \left( \mathcal{I}_{T}\left( \Omega^* \right) \right) = \mathcal{A}_T$, where
$V_T(\cdot)$ denotes the ($mr$-dimensional) volume of the set measured with respect to
the unit length $1/T$.

\smallskip

(e) By the inequality (\ref{Daven1}), rescaled with respect to the unit length
$1/T$,
\begin{multline}\label{Goat1}
\left| V_T\left( \mathcal{I}_{T}\left( \Omega^* \right) \right) -  N_T\left( \Omega^* \right) \right| \leq
O_{\Omega} \left( T^{mr-1} \right) + 
\sum_{j=0}^{mr-1}V_{T,j}\left( \mathcal{I}_{T}\left( \Omega^* \right) \right) h^{mr-1} \leq \\
O_{\Omega} \left( T^{mr-1} \right) + 
\sum_{j=0}^{mr-1}T^j V_j \left(\Omega^* \right) h^{mr-j} = O_{\Omega} \left( T^{mr-1} \right).
\end{multline}

\medskip

In the next step, we obtain a similar estimate for 
$\left| V_T\left( \mathcal{E}_{T}\left( \Omega^* \right) \right) -  N_T\left( \Omega^* \right) \right|$,
where $\mathcal{E}_{T}\left( \Omega^* \right)$ is the union of the tessellation cubes intersecting $\Omega^*$.

\smallskip

\textbf{2.} We observe the following properties of $\mathcal{E}_{T}\left( \Omega^* \right)$.

\smallskip

(a) $\Omega^* \subseteq \mathcal{E}_{T}\left( \Omega^* \right)$.

\smallskip

(b) $\mathcal{E}_{T}\left( \Omega^* \right) \in \mathcal{D}_{mr}(h)$ by Condition \ref{Peace} of Definition \ref{Defini}. 

\smallskip


(c) $N_T\left( \mathcal{E}_{T, \varepsilon}\left( \Omega^* \right) \right) = N_T\left( \Omega^* \right)$, 
by Condition \ref{Exterior} of Definition \ref{Defini}.

\smallskip

(d) $V_T\left( \mathcal{E}_{T, \varepsilon}\left( \Omega^* \right) \right) = 
T^{mr} V \left( \mathcal{E}_{T, \varepsilon}\left( \Omega^* \right) \right) \to  
T^{mr} V \left( \mathcal{E}_{T}\left( \Omega^* \right) \right) = \mathcal{B}_T$ 
as $\varepsilon \to 0$. Also $\mathcal{B}_T$ is equal to
{\em the number of cubes of (\ref{Tess}) contained
in $\mathcal{E}_{T}\left( \Omega^* \right)$}


\smallskip

(e) From $\mathcal{E}_{T}\left( \Omega^* \right) \subseteq [-1,1]^{mr}$, we conclude that there
is a positive number $\kappa$ (independent of $T$) such that
$V_j\left(\mathcal{E}_{T}\left( \Omega^* \right)  \right) \leq \kappa $
for all $j$.

\smallskip

(f) By the inequality (\ref{Daven1}), 
\begin{equation}\label{PreGoat2}
\left| V_T\left( \mathcal{E}_{T, \varepsilon}\left( \Omega^* \right) \right) -  N_T\left( \Omega^* \right) \right| \leq 
\kappa \sum_{j=0}^{mr-1}T^j  h^{mr-j}.
\end{equation}

Therefore, taking the limit as $\varepsilon \to 0$, we obtain
\begin{equation}\label{Goat2}
\left| V_T\left( \mathcal{E}_{T}\left( \Omega^* \right) \right) -  N_T\left( \Omega^* \right) \right|  = 
O_{\Omega} \left( T^{mr-1} \right).
\end{equation}

\medskip

\textbf{3.} By the triangle inequality, (\ref{Goat1}), and (\ref{Goat2}), we conclude that

\begin{equation}\label{Goat}
\left| \mathcal{A}_T  - \mathcal{B}_T \right| = O_{\Omega} \left( T^{mr-1} \right).
\end{equation}
At this point, we would like to recall that by Condition \ref{it5} of Definition \ref{Defini},
$V \left(\mathcal{I}_{T}\left( \Omega^* \right)  \right) - V \left(\Omega^*  \right) = O_{\Omega} \left( 1/T \right)$ and 
$V \left(\mathcal{E}_{T}\left( \Omega^* \right) \right) - V \left( \Omega^* \right) = O_{\Omega} \left( 1/T \right)$.
This remark together with (\ref{Goat}) finally proves (\ref{Sandwich1}).

\medskip

Next section will deal with the number $\mathfrak{N}(\textbf{J})$ of integral points $\textbf{x}$ of
$\mathcal{N}_{\mathcal{C}}$ such that $p \textbf{x}^*$ belongs to
a cube of the subdivision
(\ref{Tess}). The estimation of $\mu_{\mathcal{C}}(\Omega)$ will be done in \ref{General}.

\subsection*{\normalsize 2.4 Estimating $\mathfrak{N}(\textbf{J})$}\label{Elementary}
\addtocounter{subsection}{+1}
Let $\textbf{J}$ be a cube in the subdivision (\ref{Tess}). Then
\begin{equation}
\mathfrak{N}(\textbf{J}) = \# \{ \textbf{x} \in \mathcal{N}_{ \mathcal{C}}
\mid p \textbf{x}^*  = (x_1, x_2 - x_1, \ldots, x_{mr} - x_{mr-1}) \in \textbf{J}\}.
\end{equation}
We write the cube $\textbf{J}$ as a direct product of intervals:
\begin{equation}
\textbf{J} = \mathcal{T}_1 \times \ldots \times \mathcal{T}_{mr}.
\end{equation}
This allows us to express $\mathfrak{N}(\textbf{J})$ in terms of the characteristic
functions $\chi_{_{\mathcal{T}_j}}$ of the intervals 
$ \mathcal{T}_1, \ldots, \mathcal{T}_{mr}$ as follows:
\begin{multline}\label{Char}
\mathfrak{N}(\textbf{J}) = \sum_{\textbf{y} = (y_1, \ldots, y_{mr}) \in \textbf{J}}
\chi_{_{\mathcal{T}_1}}(y_1)\chi_{_{\mathcal{T}_2}}(y_2)
\ldots \chi_{_{\mathcal{T}_{mr}}}(y_{mr}) = \\
\{\text{recall that} ~ y_1 = x_1,~y_2 = x_2 - x_1, \ldots, y_{mr} = x_{mr} - x_{mr-1} \}\\
\sum_{\textbf{x} = (x_1, \ldots, x_{mr}) \in \mathcal{N}_{ \mathcal{C}}}
\chi_{_{\mathcal{T}_1}}(x_1)\chi_{_{\mathcal{T}_2}}(x_2 - x_1)
\ldots \chi_{_{\mathcal{T}_{mr}}}(x_{mr} - x_{mr-1}). 
\end{multline}
Now we can
write the characteristic function $\chi_{_{\mathcal{T}}}(x)$ 
as an exponential sum.
In the the next formula, we assume that $x \in \mathcal{J}$, where $\mathcal{J}$
is an interval (closed or not) of length $ \leq p$.
\begin{equation}\label{Vin}
\chi_{_{\mathcal{T}}}(x) =
p^{-1} \sum_{z \in \mathcal{T}} \sum_{k (\text{mod} p)}
e (k(x-z)/p),
\end{equation}
where the sum $\sum_{z \in \mathcal{T}}$ is taken
over the integral points of $\mathcal{T}$.
We substitute (\ref{Vin}) in (\ref{Char}), and change the order of summation:
\begin{equation}\label{Involved}
\mathfrak{N}(\textbf{J}) = p^{-mr} 
\sum_{\textbf{k} = (k_1, \ldots, k_{mr}) \in \mathbb{F}_p^{mr}} 
\prod_{j=1}^{mr}\left(\sum_{y_j \in \mathcal{T}_j} e(-k_j y_j/p)\right) S_{\mathbf{k}},
\end{equation} 
where
\begin{equation}
S_{\mathbf{k}} =  
\sum_{\mathbf{x} \in \mathcal{N}_{ \mathcal{C}}} e(\mathcal{L}_{\mathbf{k}}(\mathbf{x})/p)
\end{equation}
and, in turn,
\begin{equation}
\mathcal{L}_{\mathbf{k}}(\mathbf{x}) = 
k_{mr} x_{mr} + \sum_{s=1}^{mr-1}(k_s - k_{s+1})x_s.
\end{equation}
Since we require that $\mathcal{C}$ does not lie in a hyperplane of
$\mathbb{A}^{mr}(\bar{\mathbb{F}}_p)$, the linear form  $\mathcal{L}_{\mathbf{k}}(\mathbf{x})$
is constant on $\mathcal{C}$ if and only if
$k_1 = \ldots = k_{mr} = 0$. (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
(\ref{Involved}) with $k_1 = \ldots = k_{mr} = 0$ is 
the \textit{main term} in (\ref{Involved}). The sum of the remaining terms
will be proved to be of the lower order of magnitude, and therefore is the \textit{error term}.
We denote the main term and the error term by $M(\mathbf{J})$
and $E(\mathbf{J})$, respectively. Then
\begin{equation}\label{Ncube}
\mathfrak{N}(\mathbf{J}) = M(\mathbf{J}) + E(\mathbf{J})
\end{equation}
We will make this formula more precise below, with the final result stated in
(\ref{Cubfinerror}) of \ref{Conc}.

\subsubsection{The main term in (\ref{Ncube})}

Each $\mathcal{T}_j$ has length $p/T$ (we denote this by
$|\mathcal{T}_j| = p/T)$; therefore
\begin{equation}\label{Vol}
\text{V}(\mathbf{J}) = {\left(p/T \right)}^{mr}.
\end{equation}
By (\ref{Involved}) and (\ref{Vol}),
\begin{multline}\label{Prep}
\frac{M(\mathbf{J})}{p^{-mr} \#\mathcal{C}(\mathbb{F}_q)} = 
\prod_{j=1}^{mr} N(\mathcal{T}_j)=
\prod_{j=1}^{mr} \left(p/T + O_{m,r,d}(1)\right)=
\text{V}(\mathbf{J})\left(1 + O_{m,r,d} \left(T/p \right) \right).
\end{multline}
Hence, substituting (\ref{Vol}) in (\ref{Prep}), we have
\begin{equation}\label{PreWeil}
M(\mathbf{J}) = T^{-mr} \#\mathcal{C}(\mathbb{F}_q)
\left(1 + O_{m,r,d} \left(T/p \right) \right).
\end{equation}
At this point we further assume that
\begin{equation}\label{Assum1}
1 \leq T \leq p^{1/2}.
\end{equation}
We substitute  
(\ref{Assum1}) in (\ref{PreWeil}):
\begin{equation}\label{MainTerm}
M(\mathbf{J})  = 
T^{-mr} \#\mathcal{C}(\mathbb{F}_q)
\left(1 + O_{m,r,d} \left( p^{-1/2} \right) \right).
\end{equation}

\subsubsection{The error term in (\ref{Ncube})}

We need to estimate the error term
\begin{equation}\label{Error}
E(\textbf{J}) = p^{-mr} 
\sum_{(0, \ldots, 0) \neq (k_1, \ldots, k_{mr}) \in \mathbb{F}_p^{mr} } 
\prod_{j=1}^{mr}\left(\sum_{y_j \in \mathcal{T}_j} e 
\left( \frac{-k_j y_j}{p} \right)\right) S_{\mathbf{k}}.
\end{equation}

To estimate the sums of the form $\sum_{y_j \in \mathcal{T}_j}$ in (\ref{Error}), we need  
the following well known lemma. 
\begin{lemma}\label{Est} Let $\| \text{\textperiodcentered} \|$ denote the distance to the nearest
integer on the real line. Then for any real $a$ and integers $l \geq 1$ and $n$, we have
\begin{equation}\label{Est1}
\left| \sum_{j=n+1}^{n+l} e(j a)\right| \leq
\min \left( l, \frac{1}{2 \Vert a \Vert} \right)
\end{equation}
\end{lemma}

\begin{remark}
We assume that if $a = 0$, then
$\text{min} \left( l, 1/(2 \Vert a \Vert) \right) = l$.
\end{remark}


Now we return to the question of estimating the sums of the 
form $\sum_{y_j \in \mathcal{T}_j}$ in (\ref{Error}). By Lemma \ref{Est},

\begin{equation}\label{Trig}
\left| \sum_{y_j \in \mathcal{T}_j} e \left(-k_j y_j/p \right) \right| \leq 
\min \left\{ |\mathcal{T}_j|+1, \frac{1}{2\| k_j / p)\|} \right\}.
\end{equation}
We use the inequalities $1/x \leq 2/(x+1)$ and $1 + 1/x \leq 2$, for $ x \geq 1$, and 
substitute (\ref{Trig}) in (\ref{Error}):
\begin{multline}\label{Calc}
E(\textbf{J}) \leq 
\sum_{(0, \ldots, 0) \neq (k_1, \ldots, k_{mr}) \in \mathbb{F}_p^{mr} }
\prod_{j=1}^{mr}\frac{1}{p}\left(\min 
\left\{ p+1, \frac{p}{|k_j|}\right\} \right) \left|S_{\mathbf{k}} \right| \ll_{mr}\\
\sum_{(0, \ldots, 0) \neq (k_1, \ldots, k_{mr}) \in \mathbb{F}_p^{mr} }
\prod_{j=1}^{mr}
\left(\frac{1}{1 + |k_j|}  \right) \left|S_{\mathbf{k}}\right|.
\end{multline}
Next, for each $\textbf{x} \ne \textbf{0}$, our hypotheses
on the curve $\mathcal{C}$ (see also Remark \ref{Lou} above) allow us to apply Bombieri's
inequality (see \cite{Bombieri}, Th.~6, p.~97):
\begin{equation}\label{Bomb}
S_{\textbf{k}} = O_{m,r,d} \left(q^{1/2} \right).
\end{equation}
We substitute (\ref{Bomb}) in (\ref{Calc}):
\begin{equation}\label{Elerror}
|E(\mathbf{J})| = O_{m,r,d} \left(q^{1/2}\ln^{mr}q \right).
\end{equation}

\subsubsection{Conclusion}\label{Conc}
We recall that  Weil's theorem
states that (see also Remark \ref{Lou} above)
\begin{equation}\label{Weil}
\# \mathcal{C}(\mathbb{F}_q) = \# \mathcal{N}_{\mathcal{C}} = q + O_{m,r,d}\left(q^{1/2}\right).
\end{equation}
We substitute (\ref{MainTerm}), (\ref{Elerror}), and (\ref{Weil}) in (\ref{Ncube}):
\begin{multline}\label{PreCubfinerror}
\mathfrak{N}(\textbf{J}) =   T^{-mr} 
\left(1 + O_{m,r} \left( p^{-1/2} \right) \right)
\left(q + O_{m,r,d}\left(q^{1/2}\right) \right)
+  
O_{m,r,d} \left(q^{1/2}\ln^{mr}q \right) = \\ 
qT^{-mr}\left (1 + O_{m,r,d}\left(q^{\frac{1}{2}} \right)+ 
O_{m,r,d}\left(q^{1- \frac{1}{2m}}  \right)+ 
O_{m,r,d}\left(q^{\frac{1}{2} - \frac{1}{2m}}  \right) 
\right) + 
O_{m,r,d}\left(q^{\frac{1}{2}}\ln^{mr}q\right).
\end{multline}
At this point, we further restrict $T$ as follows:
\begin{equation}\label{Serf}
\frac{q^{1 - \frac{1}{2m}}}{T^{mr}} \leq q^{\frac{1}{2}}\ln^{mr}q.
\end{equation}
This allows us to simplify (\ref{PreCubfinerror})
in the following way:
\begin{equation}\label{Cubfinerror}
\mathfrak{N}(\textbf{J}) = q / T^{mr} + O_{m,r,d}\left(q^{1/2}\ln^{mr}q\right).
\end{equation}

\subsection*{\normalsize 2.5 Estimating $\mu_{\mathcal{C}}(\Omega)$ } \label{General}
\addtocounter{subsection}{+1}
The formula (\ref{Connection}) allows us to estimate the numerator
in (\ref{Prob})
as the product of the right-hand sides of (\ref{Sandwich1}) and (\ref{Cubfinerror}):
\begin{multline}\label{Num1}
\#\{\textbf{x} \in \mathcal{N}_{\mathcal{C}}\mid \widetilde{\textbf{x}} \in \Omega\} =
N(p \Omega^*) = \\ 
\left(q / T^{mr} + O_{m,r,d}\left(q^{1/2}\ln^{mr}q \right) \right)
\left(T^{mr} \text{V}({\Omega^*}) + O_{\Omega}\left(T^{mr-1} \right) \right) = \\
q \text{V}(\Omega^*) + O_{m,r,d,\Omega}\left(q / T \right)+
O_{m,r,d,\Omega}\left(T^{mr}q^{1/2}\ln^{mr}q \right).
\end{multline}
Next, the denominator in (\ref{Prob}) can be estimated by  (\ref{Weil}). This yields
\begin{equation}\label{Meas}
\mu_{\mathcal{C}}(\Omega) = \text{V}(\Omega^*) + 
O_{m,r,d, \Omega}\left(T^{mr} q^{-1/2} \ln^{mr}q \right) + O_{m,r,d, \Omega}(1/T).
\end{equation}
{\em Now we specify $T$ as the root of 
the equation} 
$T^{mr} q^{-1/2} \ln^{mr}q = 1/T$:
\begin{equation}\label{FinT}
T = q^{\frac{1}{2(mr+1)}} \ln^{\frac{-mr}{mr+1}}q.
\end{equation}
(We remark that (\ref{Assum1}), (\ref{Serf}), and (\ref{FinT}) agree with each other.) 
The value of $T$ given by (\ref{FinT}) allows us to have only one error term in (\ref{Meas}):
\begin{equation}\label{FinMeas}
\mu_{\mathcal{C}}(\Omega) = \text{V}(\Omega^*) + 
O_{m,r,d, \Omega} \left(q^{\frac{-1}{2(mr+1)}} \ln^{\frac{mr}{mr+1}}q \right),
\end{equation}
and we may apply (\ref{Vol1}).
This finally proves Theorem \ref{Great}.



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\end{thebibliography}



\end{document}