Group Action Approaches in Erdos Quotient Set Problem

Introduction

Let $q=p^n$ for some prime $p$ and $n \ge1$. $\mathbb{F}_q$ denotes the finite field of $q$ elements. Given $E \subset\mathbb{F}_q^d$, the Erdos Distance problem asks what that best lower bound of

is. The work of [5] formulated the problem and showed that if $|E| \ge2q^{(d+1)/2}$, then $\Delta(E)=\mathbb{F}_q$. [4] showed that the $(d+1)/2$ exponent is sharp when $d$ is odd. Moreover, if $d$ is even, it is shown in [4] that the exponent must be at least $d/2$, for the possibility of $|\Delta(E)|\gg q$. It is conjectured that if $d$ is even and $|E|\gg q^{d/2}$, then $|\Delta(E)|\gg q$. In the more general $k$-simplex setting, [1] the results of [5] are generalized bringing in group-action machinery.

There have been various special cases where the $(d+1)/2$ exponent has been improved. The current results are in the case when $d=2$. [3] proved that if $q\equiv3\pmod{4}$, $E\subset\mathbb{F}_q^2$, and $|E|\gg q^{4/3}$, then $|E|>q/2$. In the more general framework of [1], the latter result was achieved with the removal of the $q\equiv3\pmod{4}$ assumption. In [8], when $q=p$, $p$ an odd prime, the exponent has been improved to $5/4$.

A similar problem to the Erdos distance problem is the Erdos quotient problem. The Erdos quotient set is defined as

[6] formulated the Erdos quotient problem and obtained the result below using Fourier Analytic techniques.

Theorem 1.1. (Theorem 1.1 of Iosevich et al. (2019)) Let $E\subset\mathbb{F}_q^d$ $d$ even. Then if $|E|\ge9q^{d/2}$, we have

If $d \ge3$ is odd and $|E| \ge6q^{d/2}$, then

Iosevich et al. (2019) prove both their results of Theorem 1.1 are sharp in the exponent, $d$.

One drawback of the result of Theorem 1.1 is that the proof is quite sophisticated and for a fixed $r$, does not say how many distinct $(a,b,c,d) \in E^4$ such that $\lVert a-b\rVert^2/\lVert c-d\rVert^2=r$. In a restricted setting, the next result below alleviates the latter two drawbacks with a simple proof. The proof doesn’t use Fourier analysis and instead uses a group-action approach.

Theorem 1.2. (Theorem 1.2 of [9]) Let $E \subset\mathbb{F}_q^2$ with $q \equiv3 \pmod{4}$. Assume that $|E| \gg q$. Then for each $r \in(\mathbb{F}_q)^2$, the number of quadruples $(a,b,c,d) \in E^4$ such that $\lVert a-b\rVert^2/\lVert c-d\rVert^2=r$ is at least

In particular,

In this work, when $d=2$, we get the result of Theorem 1.2 for all of $\mathbb{F}_q$ in the $q \equiv3 \pmod{4}$ setting.

Recently, Iosevich et al. (2023) generalized the result of Iosevich et al. (2019) for quadratic forms. We state the result below after introducing the next definition. For $r \in\mathbb{F}_q$ define

For simplicity, we state the relevant main results of Iosevich et al. (2023) in the specific case of the quadratic form $Q(x)=\lVert x\rVert^2$.

Theorem 1.3. (Theorem 1.6 in Iosevich et al. (2023)) Assume that $E \subset\mathbb{F}_q^d$, $d \ge2$. If $d \ge2$ is even and $|E| \ge4q^{d/2}$, then

for all $r \in\mathbb{F}_q^*$.

If $d$ is odd and $|E| \ge\frac{11}{6}q^{(d+1)/2}$, then

for all $r \in\mathbb{F}_q^*$.

In Theorem 1.3, similar results to Theorem Iosevich et al. (2019) are obtained with the addition of a count, $V(r)$. In addition, like Theorem Iosevich et al. (2019), the proof of Theorem 1.3 is quite sophisticated.

The construction of the matrices $A_{even}$ and $A_{odd}$ from the proof of Theorem 2.1 allows us to improve the main result of Bhowmik & Rakhmonov (2023), which we state below. Specifically, we allow for any $r \in\mathbb{F}_q^d$ instead of just $r \in(\mathbb{F}_q^2)$.

Theorem 1.4. (Theorem 1.3 [2]) Suppose $r \in(\mathbb{F}_q^2) \setminus\{0\}$ and $\varnothing\ne A \subset\{(i,j):1 \le i < j \le k+1\}$ where $k \ge1$. If $E \subset\mathbb{F}_q^d$ with $|E| \ge2kq^{d/2}$, then there exist $(x_1,\ldots,x_{k+1}),(y_1,\ldots,y_{k+1}) \in E^{k+1}$ such that $\lVert y_i-y_j\rVert^2=r\lVert x_i-x_j\rVert^2$ if $(i,j) \in A$ and $x_i \ne x_j$, $y_i \ne y_j$ if $1 \le i < j \le k+1$.

Results

We use the group action approach and remove the restriction of $r \in(\mathbb{F}_q)^2$ from Theorem 1.2, allowing for any $r \in\mathbb{F}_q$.

Theorem 2.1. Let $q$ be a prime power and $E \subset\mathbb{F}_q^d$. If $q \equiv3 \pmod{4}$, then for each $r \in\mathbb{F}_q^*$,

if $|E| \ge\sqrt{2}q$.

The results of Theorem 2.1 are sharp. For example, in section 1.1 of [6], if $q=p^2$, in the $d=2$ setting, the set

has size $q$, but

not $p^2$.

Using the proof techniques of Theorem 2.1, we improve the main result of [2], by allowing for any $r \in\mathbb{F}_q^d$ instead of just $r \in(\mathbb{F}_q^2)$.

Theorem 2.2. Suppose $r \in(\mathbb{F}_q)\setminus\{0\}$ and $\varnothing\ne A \subset\{(i,j):1 \le i < j \le k+1\}$ where $k \ge1$. If $E \subset\mathbb{F}_q^d$ with $|E| \ge2kq^{d/2}$, if $d$ is even or $|E| \ge2kq^{(d+1)/2}$, if $d$ is odd, then there exist $(x_1,\ldots,x_{k+1}),(y_1,\ldots,y_{k+1}) \in E^{k+1}$ such that $\lVert y_i-y_j\rVert^2=r\lVert x_i-x_j\rVert^2$ if $(i,j)\in A$ and $x_i\ne x_j$, $y_i\ne y_j$ if $1\le i<j\le k+1$.

Note the results are of Theorem 2.2 are sharp. In the case where $q=p^{2l}$, $p\equiv3\pmod{4}$, and $l\equiv1\pmod{2}$ the exponent $d/2$ is sharp (See section 6, page 9, [2]).

Proof. Proof of Theorem 2.2. The proof follows almost verbatim of the combinatorial proof of Theorem 1.3 of [2] in the $r\ne0,1$ case. We assume that $r\ne1$ is in $\mathbb{F}_q^*$. Recall the matrix, $A$, from the proof Theorem 2.1. In the proof of Theorem 1.3 of 2.2, page 8, we replace the set $tE$ with

where $A=\sqrt{r}A_{odd}$, if $d$ is odd, and $A=\sqrt{r}A_{even}$, if $d$ is even, which we define below.

Since $d$ is even, we can choose $A$ to have $\frac{d}{2}$ blocks. Note that $A_{even}$ is orthogonal by construction. Denote $O(d)$, the group of orthogonal matrices in $\mathbb{F}_q^d$. If $d\ge3$ is odd we use the block matrix

noting $A_{odd}$ is orthogonal by construction.

As in the proof of Theorem 1.3 of 2.2, we define

if $d$ is even. If $d$ is odd we set

If $d$ is even, then

by the averaging argument of Theorem 1.3 of 2.2. Similarly if $d$ is odd,

We get $\{x_i\}_{i=1}^{k+2} \subset AE \cap(E+a)$ for some $a \in\mathbb{F}_q^{d-1} \times\mathbb{F}_{q^2}$ in the odd $d$ case and $a \in\mathbb{F}_q^d$ in the even $d$ case. Thus

and

for all $i=0,\ldots,k+2$.

Thus

where the last equality above comes from the fact that $\sqrt{r}A$ is orthogonal.

Proof. Proof of Theorem 2.1. The proof is similar to Theorem 1.2. Assume that $r \in\mathbb{F}_q^*$. Then $\sqrt{r} \in\mathbb{F}_{q^2}$. For any characteristic, $p$, it is a basic fact from the theory of finite fields that there is some $a,b \in\mathbb{F}_q$ such that $a^2+b^2=r$. We construct the diagonal matrix,

Note that $A$ is orthogonal by construction. Denote $O(2)$, the group of orthogonal matrices in $\mathbb{F}_q^2$.

Let $z \in\mathbb{F}_q^2$ and $\theta\in O(2)$. Define the set,

Consider the sum,

By the Cauchy Schwarz inequality,

It follows that,

$\sum_{\theta,z}\eta_\theta(z)^2$ counts the number of $(u,v,w,x)\in E^4$ such that $\|u-v\|^2=r\|w-x\|^2$ modulo double counting by the stabilizer of $c-d$ for each $(c,d)\in E^2$. The stabilizer of each $(c,d)\in E^2\backslash\operatorname{diag}(E)$ has $|O(1)|$ elements (see page 3 of [1]).

The number of tuples, $(u,v,w,x)$ in the sum, $\sum_{\theta,z}\eta_\theta(z)^2$, where $u=v$ and $w=x$ are counted $\sum_{\theta\in O(2),z\in\mathbb{F}_q^2}\eta_\theta(z)=|E|^2|O(2)|$ times.

Define

Since $q \equiv3 \pmod{4}$, $N_0 = 0$. Thus, if $\lvert E\rvert\ge\sqrt{2q}$,

From above, if $\lvert E\rvert\ge\sqrt{2q}$, then $V(r) \ge\frac{\lvert E\rvert^4(q+1)}{4q^2}$, and we’re done.