Extension theorems for Hamming varieties over finite fields

TL;DR

This paper establishes Stein-Tomas extension estimates for Hamming varieties over finite fields, despite poor Fourier decay, advancing understanding of harmonic analysis in finite field settings.

Contribution

It proves the Stein-Tomas $L^2\to L^r$ extension estimate for Hamming varieties over finite fields, a novel result given their Fourier decay properties.

Findings

Stein-Tomas extension estimate holds for Hamming varieties

Fourier decay bound is not optimal but extension estimate still valid

Advances finite field harmonic analysis understanding

Abstract

We study the finite field extension estimates for Hamming varieties $H_j, j\in \mathbb F_q^*,$ defined by $H_j=\{x\in \mathbb F_q^d: \prod_{k=1}^d x_k=j\},$ where $\mathbb F_q^d$ denotes the $d$-dimensional vector space over a finite field $\mathbb F_q$ with $q$ elements. We show that although the maximal Fourier decay bound on $H_j$ away from the origin is not good, the Stein-Tomas $L^2\to L^r$ extension estimate for $H_j$ holds.

Equations

$$\Vert g{\Vert }_{{\ell }^s({\mathbb{F}}_q^d)}:={\left(\sum _{m\in {\mathbb{F}}_q^d}|g(m){|}^s\right)}^{1/s}\text{mbox}and\Vert f{\Vert }_{L^s({\mathbb{F}}_q^d)}:={\left(q^{-d}\sum _{x\in {\mathbb{F}}_q^d}|f(x){|}^s\right)}^{1/s}.$$

$$\Vert f{\Vert }_{L^s(V,d\sigma )}:={\left(\frac{1}{|V|}\sum _{x\in V}|f(x){|}^s\right)}^{1/s}.$$

$$\hat{g}(x)=\sum _{m\in {\mathbb{F}}_q^d}\chi (-m\cdot x)g(m),$$

$$\sum_{\alpha\in\mathbb{F}_{q}^{d}}\chi(n\cdot\alpha)=\left\{\begin{array}[]{ll}0&\mbox{if}\quad n\neq(0,\ldots,0)\\ q^{d}&\mbox{if}\quad n=(0,\ldots,0).\end{array}\right.$$

$$f^{\vee }(m):=q^{-d}\sum _{x\in {\mathbb{F}}_q^d}\chi (m\cdot x)f(x).$$

$$(fd\sigma {)}^{\vee }(m):=\frac{1}{|V|}\sum _{x\in V}\chi (m\cdot x)f(x).$$

$$\Vert (fd\sigma {)}^{\vee }{\Vert }_{{\ell }^r({\mathbb{F}}_q^d)}\le R^{\ast }(p\to r)\Vert f{\Vert }_{L^p(V,d\sigma )}$$

$$\Vert \hat{g}{\Vert }_{L^{p^{\prime }}(V,d\sigma )}\le R_V^{\ast }(p\to r)\Vert g{\Vert }_{{\ell }^{r^{\prime }}({\mathbb{F}}_q^d)},$$

$$r\ge \frac{2d}{d-1}\text{mbox}andr\ge \frac{p(d-k)}{(p-1)(d-1-k)}.$$

$$R_V^{\ast }(2\to r)\lesssim 1\text{mbox}if\frac{2(d-k)}{d-k-1}\le r\le \infty .$$

$$R_V^{\ast }\left(2\to \frac{2(d-k)}{d-k-1}\right)\lesssim 1.$$

$$|V|\sim q^{d-1}$$

$$\underset{m\in {\mathbb{F}}_q^d\setminus \{(0,\dots ,0)\}}{\max }|(d\sigma {)}^{\vee }(m)|\lesssim q^{-\frac{\alpha }{2}}$$

$$R_V^{\ast }\left(2\to \frac{2(\alpha +2)}{\alpha }\right)\lesssim 1.$$

$$\underset{m\in {\mathbb{F}}_q^d\setminus \{(0,\dots ,0)\}}{\max }|(d\sigma {)}^{\vee }(m)|\sim q^{-\frac{(d-2)}{2}}$$

$$H_j=\left\{x=(x_1,\dots ,x_d)\in {\mathbb{F}}_q^d:\prod _{k=1}^d{x}_k=j\right\}.$$

$$R_{H_j}^{\ast }\left(2\to \frac{2d+2}{d-1}\right)\lesssim 1.$$

$$R_{H_j}^{\ast }(2\to r)\lesssim 1\text{mbox}if\frac{2d}{d-1}\le r\le \infty .$$

$$\underset{m\in {\mathbb{F}}_q^d\setminus \{(0,\dots ,0)\}}{\max }|(d{\sigma }_j{)}^{\vee }(m)|\sim q^{-1}\text{mbox}ford\ge 4.$$

$$\Lambda (E)=\sum _{x,y,z,w\in E:x+y=z+w}1.$$

$$\Vert g{\Vert }_{{\ell }^{p_2}({\mathbb{F}}_q^d)}\le \Vert g{\Vert }_{{\ell }^{p_1}({\mathbb{F}}_q^d)}$$

$$\Vert f{\Vert }_{L^{p_1}({\mathbb{F}}_q)}\le \Vert f{\Vert }_{L^{p_2}({\mathbb{F}}_q^d)},\Vert f{\Vert }_{L^{p_1}(V,d\sigma )}\le \Vert f{\Vert }_{L^{p_2}(V,d\sigma )}.$$

$$\Vert \hat{g}{\Vert }_{L^2({\mathbb{F}}_q^d)}=\Vert g{\Vert }_{{\ell }^2({\mathbb{F}}_q^d)}\text{mbox}or\Vert f^{\vee }{\Vert }_{{\ell }^2({\mathbb{F}}_q^d)}=\Vert f{\Vert }_{L^2({\mathbb{F}}_q^d)}$$

$$g_1\ast g_2(m)=\sum _{n\in {\mathbb{F}}_q^d}{g}_1(m-n)g_2(n).$$

$$\Vert g_1\ast g_2{\Vert }_{{\ell }^r({\mathbb{F}}_q^d)}\le \Vert g_1{\Vert }_{{\ell }^a({\mathbb{F}}_q^d)}\Vert g_2{\Vert }_{{\ell }^b({\mathbb{F}}_q^d)}.$$

$$\Vert Tg{\Vert }_{{\ell }^{r_0}({\mathbb{F}}_q^d)}\le M_0\Vert g{\Vert }_{{\ell }^{p_0}({\mathbb{F}}_q^d)}\text{mbox}and\Vert Tg{\Vert }_{{\ell }^{r_1}({\mathbb{F}}_q^d)}\le M_1\Vert g{\Vert }_{{\ell }^{p_1}({\mathbb{F}}_q^d)}.$$

$$\Vert Tg{\Vert }_{{\ell }^r({\mathbb{F}}_q^d)}\le M_0^{1-\theta }{M}_1^{\theta }\Vert g{\Vert }_{{\ell }^p({\mathbb{F}}_q^d)}$$

$$\frac{1-\theta }{r_0}+\frac{\theta }{r_1}=\frac{1}{r}\text{mbox}and\frac{1-\theta }{p_0}+\frac{\theta }{p_1}=\frac{1}{p}.$$

$$(d{\sigma }_j{)}^{\vee }(m)=(-1{)}^{d-{\ell }_m}(q-1{)}^{-(d-{\ell }_m)}\text{mbox}if1\le {\ell }_m\le d.$$

$$(d{\sigma }_j{)}^{\vee }(m)=\frac{1}{|H_j|}\sum _{x\in H_j}\chi (m\cdot x)=\frac{1}{|H_j|}\sum _{x_1,x_2,\dots ,x_d\in {\mathbb{F}}_q^{\ast }:x_1{x}_2\cdots x_d=j}\chi (m\cdot x).$$

Full text

Key words and phrases: Finite field, Hamming variety, Extension operator

The first and second listed authors were supported by Basic Science Research Programs through National Research Foundation of Korea (NRF) funded by the Ministry of Education (NRF-2018R1D1A3B07045594 and NRF-2018R1D1A1B07044469, respectively). The third listed author was supported by Swiss National Science Foundation grant P400P2-183916..

1. Introduction

The extension or restriction problem is one of central open questions in Euclidean harmonic analysis. In 2002, Mockenhaupt and Tao [16] initially studied this problem for algebraic varieties in the finite field setting. Let $\mathbb{F}_{q}^{d}$ be the $d$-dimensional vector space over a finite field with $q$ elements. Throughout this paper, we assume that $q$ is an odd prime power. Given complex-valued functions $f,g$ on $\mathbb{F}_{q}^{d}$ and $1\leq s<\infty,$ we define

[TABLE]

In addition, it is defined that $\|g\|_{\ell^{\infty}(\mathbb{F}_{q}^{d})}:=\max_{m\in\mathbb{F}_{q}^{d}}|g(m)|$ and $\|f\|_{L^{\infty}(\mathbb{F}_{q}^{d})}:=\max_{x\in\mathbb{F}_{q}^{d}}|f(x)|.$ The notation $\|g\|_{\ell^{s}(\mathbb{F}_{q}^{d})}$ indicates that the function $g$ is defined on the space $\mathbb{F}_{q}^{d}$ with counting measure. On the other hand, the notation $\|f\|_{L^{s}(\mathbb{F}_{q}^{d})}$ tells us that the function $f$ is defined on the space $\mathbb{F}_{q}^{d}$ with normalized counting measure. Let $V$ be an algebraic variety in $\mathbb{F}_{q}^{d}.$ We endow $V$ with a normalized surface measure $d\sigma$ which means that the mass of each point of $V$ is $1/|V|,$ where $|V|$ denotes the cardinality of the set $V.$ For a function $f:V\to\mathbb{C}$ and $1\leq s<\infty,$ we define

[TABLE]

We also define $\|f\|_{L^{\infty}(V,d\sigma)}:=\max_{x\in V}|f(x)|.$

The Fourier transform of $g$, denoted by $\widehat{g},$ is defined by

[TABLE]

where $\chi$ denotes the canonical additive character of $\mathbb{F}_{q}$, and $m\cdot x$ is the usual dot-product of $m$ and $x.$ We recall that the orthogonality of $\chi$ states that

[TABLE]

The inverse Fourier transform of $f$, denoted by $f^{\vee}$, is defined by

[TABLE]

Furthermore, the inverse Fourier transform of the measure $fd\sigma$ is given by

[TABLE]

We denote by $R^{*}(p\to r)$ the smallest constant such that the following extension estimate

[TABLE]

holds for all functions $f$ on $V.$ Note that $R_{V}^{*}(p\to r)$ may depend on $q$, the size of the underlying finite field $\mathbb{F}_{q}.$ The extension problem for the variety $V$ is to determine all exponents $1\leq p,r\leq\infty$ such that $R_{V}^{*}(p\to r)$ is independent of $q.$ For $A,B>0$, we will write $A\lesssim B$ if $A\leq CB$ for some constant $C>0$ independent of $q.$ We will also use $A\sim B$ if $A\lesssim B$ and $B\lesssim A.$ By a well-known duality, the inequality (1.1) is the same as the following restriction estimate:

[TABLE]

where $p^{\prime},r^{\prime}$ denote the dual exponents of $p,r$, respectively (i.e. $1/p+1/p^{\prime}=1$ and $1/r+1/r^{\prime}=1$).

When $|V|\sim q^{d-1}$, necessary conditions for $R^{*}_{V}(p\to r)$ bound can be obtained from the size of a maximal affine subspace lying on $V.$ Indeed, Mockenhaupt and Tao [16] showed that if $|V|\sim q^{d-1}$ and $V$ contains an affine subspace $H$ with $|H|=q^{k},$ then necessary conditions for $R^{*}_{V}(p\to r)$ bound are given by

[TABLE]

In dimension two, the extension problem for algebraic curves $V$ was completely solved by Shen and the second listed author [10] who showed that the above necessary conditions are also sufficient conditions for $R^{*}_{V}(p\to r)$ bound. For this reason, we will restrict ourselves to the case when $d\geq 3.$ In particular, we have the following conjecture for $R^{*}_{V}(2\to r)$ bound.

Conjecture 1.1**.**

Let $V$ be an algebraic variety in $\mathbb{F}_{q}^{d}.$ Suppose that $|V|\sim q^{d-1}$ and $V$ contains an affine subspace $H$ with $|H|=q^{k}.$ Then we have

[TABLE]

By the norm nesting property (see Section 2), one can check that if $1\leq r_{1}\leq r_{2}\leq\infty,$ then $R^{*}_{V}(2\to r_{2})\leq R^{*}_{V}(2\to r_{1}).$ This implies that a smaller exponent gives a better result on the restriction problem. Thus if we want to establish the sharp $R_{V}^{*}(2\to r)$ bound, then we only needs to find the smallest exponent $r$ such that $R_{V}^{*}(2\to r)\lesssim 1.$ Namely, to confirm Conjecture 1.1 it suffices to prove that

[TABLE]

The finite field extension problem has been studied only for few algebraic varieties with relatively simple structures such as spheres, paraboloids, or cones. For example, Mockenhaupt and Tao [16] addressed results on the problem for paraboloids and cones, and their work for those varieties has been recently improved by other researchers (see [11, 4, 14, 12, 8, 6, 17, 9, 13]). For spheres, Iosevich and the second listed author [5] obtained nontrivial results which have been improved in the papers [7, 9]. While several new methods have been used in studying the Euclidean extension problem, there are only few known skills to deduce the results on the finite field extension problem. Among other things, the Stein-Tomas argument can be applied in the finite field case to deduce $R^{*}_{V}(2\to r)$ bound. Indeed, Mockenhaupt and Tao [16] introduced the finite field Stein-Tomas argument. In particular, we have the following lemma which is a special case of Lemma 6.1 in [16].

Lemma 1.2** (The finite field Stein-Tomas argument).**

Let $d\sigma$ be the normalized surface measure on an algebraic variety $V$ in $\mathbb{F}_{q}^{d}.$ Suppose that

[TABLE]

and

[TABLE]

for some $\alpha>0.$ Then we have

[TABLE]

For a general version of Lemma 1.2, we refer readers to [1]. To apply Lemma 1.2, one needs to compute the maximal Fourier decay bound on the measure $d\sigma$ away from the origin. For example, when $V$ is a sphere or a paraboloid, it is well-known that one can take $\alpha=(d-1)/2$ in (1.3), and thus $R^{*}_{V}(2\to(2d+2)/(d-1))\lesssim 1$ (see [16, 5]). This result is called as the Stein-Tomas result which gives the optimal $R^{*}_{V}(2\to r)$ bound in general. Now, we pose an interesting question.

Question 1.3**.**

Does there exist a variety $V$ such that the condition (1.3) does not hold with $\alpha=(d-1)/2$ but we still have the Stein-Tomas result for $V$?

There exist several varieties $V$ in $\mathbb{F}_{q}^{d}$ such that the condition (1.3) does not hold with $\alpha=(d-1)/2$ and the Stein-Tomas result can not be obtained. For example, if $d$ is even and $d\sigma$ is the normalized surface measure on the variety $V:=\{x\in\mathbb{F}_{q}^{d}:x_{1}^{2}-x_{2}^{2}+\cdots+x_{d-1}^{2}-x_{d}^{2}=0\}$, then

[TABLE]

and the optimal $L^{2}\to L^{r}$ extension estimate for $V$ is that $R^{*}_{V}(2\to 2d/(d-2))\lesssim 1,$ which is much weaker than the Stein-Tomas result (see Theorem 2.1 and Lemma 4.1 in [11]).

Our main purpose of this paper is to provide a concrete variety which gives a positive answer to the above question.

For each $j\in\mathbb{F}_{q}^{*}$, the Hamming variety $H_{j}$ in $\mathbb{F}_{q}^{d}$ is defined by

[TABLE]

Since $j\neq 0$, it is not hard to see $|H_{j}|=(q-1)^{d-1}\sim q^{d-1}.$ Our main result is as follows.

Theorem 1.4**.**

Let $d\sigma_{j}$ denote the normalized surface measure on the Hamming variety $H_{j}$ in $\mathbb{F}_{q}^{d}$ defined as in (1.4). Then, for every $j\neq 0$ and $d\geq 3,$ we have

[TABLE]

It is not hard to see that the Hamming variety $H_{j}$ with $j\neq 0$ does not contain any line. Taking $k=0$ in Conjecture 1.1 we may conjecture that

[TABLE]

Theorem 1.4 is much weaker than this conjecture, but for $d\geq 4$ it can not be obtained by simply applying Lemma 1.2. To see this, notice from Corollary 3.3 in Section 3 that

[TABLE]

Combining this with Lemma 1.2, we only see that $R^{*}_{H_{j}}(2\to 4)\lesssim 1$ which is much weaker than Theorem 1.4 for $d\geq 4.$ To prove Theorem 1.4, we decompose the surface measure on $H_{j}$ into $(d+1)$ surface measures and each of them will be analyzed.

Remark 1.5*.*

In even dimensions, much progress on extension problems for the paraboloid $P$ in $\mathbb{F}_{q}^{d}$ has been made by improving the additive energy estimate for subsets of the paraboloid (see, for example, [12, 6, 17]). Recall that for a set $E$ in $\mathbb{F}_{q}^{d}$, the additive energy of the set $E$, denoted by $\Lambda(E)$, is defined by

[TABLE]

When a set $E$ lies on the Hamming variety $H_{j}$, it seems that it is a challenging problem to obtain a good upper bound of $\Lambda(E).$

2. Discrete Fourier analysis

In this section, we review the discrete Fourier analysis which will be our main tool in proving our main result. The proofs of all statements in this section can be found in Been Green’s lecture note [3]. In the finite field setting, the norm nesting properties hold: for $1\leq p_{1}\leq p_{2}\leq\infty,$

[TABLE]

and

[TABLE]

The Plancherel theorem states that

[TABLE]

which can be easily deduced by the orthogonality of $\chi.$ We also note that $\widehat{(f^{\vee})}=f.$ Given functions $g_{1},g_{2}:\mathbb{F}_{q}^{d}\to\mathbb{C},$ the convolution function of $g_{1}$ and $g_{2}$, denoted by $g_{1}\ast g_{2}$, is defined by

[TABLE]

One can easily check that $\widehat{g_{1}\ast g_{2}}=\widehat{g_{1}}\widehat{g_{2}}.$ We recall that Young’s inequality for convolutions states that if $1\leq a,b,r\leq\infty$ satisfy $1/r=1/a+1/b-1,$ then

[TABLE]

We will invoke the following well-known interpolation theorem.

Theorem 2.1** (Riesz-Thorin).**

Let $1\leq p_{0},p_{1},r_{0},r_{1}\leq\infty$ with $p_{0}\leq p_{1}$ and $r_{0}\leq r_{1}.$ Suppose that $T$ is a linear operator and the following two estimates hold for all functions $g$ on $\mathbb{F}_{q}^{d}:$

[TABLE]

Then we have

[TABLE]

for any $0\leq\theta\leq 1$ with

[TABLE]

3. Fourier decay on Hamming varieties

Recall that $d\sigma_{j}$ denotes the normalized surface measure on the Hamming variety $H_{j}$ in $\mathbb{F}_{q}^{d}.$ In this section, we introduce an explicit form of $(d\sigma_{j})^{\vee}$ which makes a crucial role in proving Theorem 1.4.

Lemma 3.1**.**

For each $j\in\mathbb{F}_{q}^{*}$, let $d\sigma_{j}$ be the normalized surface measure on the Hamming variety $H_{j}$ in $\mathbb{F}_{q}^{d}.$ For each $m\in\mathbb{F}_{q}^{d},$ denote by $\ell_{m}$ the number of zero components of $m.$ Then we have

[TABLE]

In addition, if $\ell_{m}=0$, then $|(d\sigma_{j})^{\vee}(m)|\lesssim q^{-\frac{(d-1)}{2}}.$

Proof.

Since $j\neq 0,$ we see that $|H_{j}|=(q-1)^{d-1}\sim q^{d-1}$ and all components of any element in the Hamming variety $H_{j}$ are not zero. By definition, it follows

[TABLE]

Case 1. Assume that $\ell_{m}=d.$ Then $m=(0,\ldots,0)$ and so $(d\sigma)^{\vee}(m)=1.$

Case 2. Assume that $\ell_{m}=k$ for some $k=1,2,\ldots,(d-1).$ Without loss of generality, we may assume that $m_{1}=m_{2}=\cdots=m_{k}=0,$ and $m_{i}\neq 0$ for $k+1\leq i\leq d.$ It follows that

[TABLE]

where we assume that if $k=d-1,$ then $\chi(m_{k+1}x_{k+1}+\cdots+m_{d-1}x_{d-1})=1.$ Since $j,m_{d}\neq 0$, we see from the orthogonality of $\chi$ that for each $x_{2},\cdots,x_{d-1}\in\mathbb{F}_{q}^{*},$

[TABLE]

Therefore we have

[TABLE]

Since $|H_{j}|=(q-1)^{d-1}$ and $m_{k+1},\ldots,m_{d-1}\neq 0$, we conclude from the orthogonality of $\chi$ that

[TABLE]

which completes the proof in the case when $1\leq\ell_{m}\leq(d-1).$

Case 3. Assume that $\ell_{m}=0.$ Then all components of $m$ are not zero. As in Case 2, we can write

[TABLE]

Since $|H_{j}|\sim q^{d-1},$ the last part of the theorem is a direct consequence from the following theorem due to Deligne [2]:

Theorem 3.2** (Multiple Kloosterman sums).**

For $a_{1},a_{2},\ldots,a_{s},b\in\mathbb{F}_{q}^{*}$, we have

[TABLE]

To find further references for Multiple Kloosterman sums, we refer readers to [P.254, [15]]. ∎

The following result follows immediately from Lemma 3.1.

Corollary 3.3**.**

For each $j\in\mathbb{F}_{q}^{*}$, let $d\sigma_{j}$ denote the normalized surface measure on $H_{j}$ in $\mathbb{F}_{q}^{d}.$ Then we have

[TABLE]

4. Proof of Theorem 1.4

We aim to prove that the extension estimate

[TABLE]

holds for all complex-valued functions $f$ on $H_{j}.$ By duality, it suffices to prove that the restriction estimate

[TABLE]

holds for all complex-valued functions $g$ on $\mathbb{F}_{q}^{d}.$ By the $RR^{*}$ method (see [3]), we see that

[TABLE]

Here, we recall that if $g,h:\mathbb{F}_{q}^{d}\to\mathbb{C},$ then

[TABLE]

For each $k=0,1,\ldots,d,$ define

[TABLE]

We decompose $(d\sigma_{j})^{\vee}$ as

[TABLE]

It follows that

[TABLE]

Hence, to complete the proof, it will be enough to show that the following two inequalities hold for all functions $g:\mathbb{F}_{q}^{d}\to\mathbb{C}$ and for all $k=1,2,\ldots,d:$

[TABLE]

and

[TABLE]

In the following subsections, we will give the proofs of inequalities (4.1) and (4.2), which completes the proof of Theorem 1.4.

4.1. Proof of inequality (4.1)

By Hölder’s inequality, we have

[TABLE]

Thus, in order to prove the inequality (4.1), it is enough to prove the following:

[TABLE]

By the Riesz-Thorin interpolation theorem (Theorem 2.1), it suffices to prove the following two inequalities:

[TABLE]

and

[TABLE]

For the first inequality (4.3), using the Plancherel theorem gives us the following estimate:

[TABLE]

Now, for each fixed $x\in\mathbb{F}_{q}^{d},$ we have

[TABLE]

Using a change of variable by letting $z=y-x,$ and the triangle inequality,

[TABLE]

We decompose $\sum_{z\in\mathbb{F}_{q}^{d}}$ into $\sum_{k=0}^{d}\sum_{z\in N_{k}}$, and use the orthogonality of $\chi.$ Then we obtain

[TABLE]

where we also used the fact that $|N_{k}|={d\choose k}q^{d-k}\sim q^{d-k}.$ Thus the inequality (4.3) holds.

The second inequality (4.4) follows by using Young’s inequality for convolutions and the second part of Lemma 3.1. More precisely, we have

[TABLE]

Hence, the proof of the inequality (4.1) is complete.

4.2. Proof of inequality (4.2)

We will prove much better inequality than the inequality (4.2). Notice from the norm nesting property (2.1) that

[TABLE]

To complete the proof of the inequality (4.2), it will be enough to show that for each $k=1,2,\ldots,d,$

[TABLE]

By Hölder’s inequality, we have

[TABLE]

As seen in (4.5), it suffices by the Plancherel theorem to prove that

[TABLE]

Fix $x\in\mathbb{F}_{q}^{d}$ and $k\in\{1,2,\ldots,d\}.$ From (3.1) of Lemma 3.1, we see that for each $m\in\mathbb{F}_{q}^{d},$

[TABLE]

Therefore, we have

[TABLE]

Since $|\widehat{N_{k}}(x)|=|\sum_{m\in N_{k}}\chi(-m\cdot x)|\leq|N_{k}|={d\choose{k}}q^{d-k}\sim q^{d-k},$ we conclude that

[TABLE]

Thus, the inequality (4.7) holds, which completes the proof of the inequality (4.2).