Pinned Distances in Modules over Finite Valuation Rings

TL;DR

This paper establishes lower bounds on the number of distinct distances from a fixed point in the Cartesian product of a subset of a finite valuation ring, showing that large subsets determine many distances.

Contribution

It proves a new lower bound on pinned distances in modules over finite valuation rings, extending distance problems to this algebraic setting.

Findings

Existence of a point with many pinned distances

If |A| ≥ q^{r - 1/3}, A×A determines a positive proportion of all distances

Lower bounds depend on the size of A and the ring's parameters

Abstract

Let $R$ be a finite valuation ring of order $q^r$ where $q$ is odd and $A$ be a subset of $R$. In the present paper, we prove that there exists a point $u$ in the Cartesian product set $A\times A\subset R^2$ such that the size of the pinned distance set at $u$ satisfies $$|\Delta_u(A\times A)|\gg \min\left\{q^r, \frac{|A|^3}{q^{2r-1}}\right\}.$$ This implies that if $|A|\ge q^{r-\frac{1}{3}}$, then the set $A\times A$ determines a positive proportion of all possible distances.

Equations

$$|{\Delta }_u(A\times A)|\gg \min \left\{q^r,\frac{|A{|}^3}{q^{2r-1}}\right\}.$$

$$|{\Delta }_{(a,b)}(A\times A)|=\Omega (\min \{p,|A{|}^{3/2}\}).$$

$$|E|\gg l(l+1)q^{\frac{(2l-1)}{2l}d+\frac{1}{2l}}.$$

$$\Delta (E)\supset {\mathbb{Z}}_q^{\ast }\cup \{0\}.$$

$$|{\Delta }_{{\mathbb{Z}}_q}(A^n)|\gg \min \left\{q,\frac{|A{|}^{2n-1}}{(r{q}^{2-\frac{1}{r}}{)}^{n-1}}\right\}.$$

$$R\supset (\pi )\supset ({\pi }^2)\cdots \supset ({\pi }^r)=0,$$

$${\mathbb{Z}}_{p^k}\supset (p)\supset (p^2)\supset \cdots \supset (p^k)=0.$$

$$||u-v||=(u_1-v_1{)}^2+\cdots +(u_d-v_d{)}^2.$$

$$\Delta (E)=\{||u-v||:u,v\in E\},$$

$${\Delta }_u(E)=\{||u-v||:v\in E\}.$$

$$|{\Delta }_u(A\times A)|\gg \min \left\{q^r,\frac{|A{|}^3}{q^{2r-1}}\right\}.$$

$$|{\Delta }_u(A\times A)|\gg \min \left\{p,\frac{|A{|}^3}{p}\right\}.$$

$$\Delta (E)\supset {\mathbb{Z}}_q^{\ast }.$$

$$|{\Delta }_{{\mathbb{Z}}_q}(A\times A)|\gtrsim \min \left\{q,\frac{|A{|}^3}{r{q}^{2-1/r}}\right\}$$

$$2u\cdot (v-w)+||w||-||v||=0,$$

$$I_w=\sum _{q\in Q,\pi \in \Pi }w(q)w(\pi ){\delta }_{q\pi },$$

$$\delta_{q\pi}=\left\{\begin{array}[]{rl}1&\text{if }q\in\pi,\\ 0&\text{if }q\notin\pi.\end{array}\right.$$

$$I_w=\sum _{q\in \pi }w(q)w(\pi )\ll \frac{1}{q^r}{W}^2+q^{2r-1}W.$$

$$2u_1(v_1-w_1)+2u_2(v_2-w_2)+(w_2^2-v_2^2)=v_1^2-w_1^2$$

$$(2u_1,v_2-w_2,w_2^2-v_2^2)\cdot (v_1-w_1,2u_2,1)=v_1^2-w_1^2.$$

$$Q=\{(2u_1,v_2-w_2,w_2^2-v_2^2):u_1,v_2,w_2\in A\}$$

$$\Pi =\{\{x\in R^3:x\cdot (v_1-w_1,2u_2,1)=v_1^2-w_1^2\}:v_1,w_1,u_2\in A\}$$

$$N$$

$$|{\Delta }_u(A\times A)|\ge \frac{|A{|}^6}{N}\gg \min \left\{q^r,\frac{|A{|}^3}{q^{2r-1}}\right\}$$

Full text

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This implies that if $|A|\geq q^{r-\frac{1}{3}}$, then the set $A\times A$ determines a positive proportion of all possible distances.

1 Introduction

Erdős-Falconer type problems in discrete geometry ask for a threshold on the size of a set so that the set determines the given geometric configurations. These problems have been studied by many authors both in continuous and discrete setting.

In [9], Erdős observed that $\sqrt{n}\times\sqrt{n}$ integer grid determines $C(\frac{n}{\sqrt{logn}})$ distances and he conjectured that the minimum number of distances determined by a $n$-point set in the plane is indeed $C(\frac{n}{\sqrt{logn}})$, where $C$ is an absolute constant. Despite many works and progress, the Erdős distance problem was open until recently. In 2010, Guth and Katz [10] employed a polynomial partitioning technique based on Elekes-Sharir framework to prove that $n$ points in the plane determine at least $C\frac{n}{logn}$ distances. This result solved the Erdős distance problem up to a $\sqrt{\log}$ factor.

The distance problem in finite field plane was first studied by Bourgain, Katz and Tao in [5]. The result in [5] was later generalized by various authors to higher dimensional vector spaces. It was also extended to many other geometric configurations in finite field geometry, see for instance [2, 3, 4, 6, 7, 11, 13, 14, 16] and references therein. In particular, in [16], Petridis proved the following pinned distance result for Cartesian product subsets of vector spaces over prime fields.

Theorem 1.1**.**

[16, Theorem 1.1]* Let $p$ be an odd prime and $A\subset\mathbb{F}_{p}$. There exist $a,b\in A$ such that*

[TABLE]

Similar geometric problems in modules over finite cyclic rings were studied by Covert, Iosevich and Pakianathan in [8]. Using a Fourier analytic approach, the authors of [8] proved the following.

Theorem 1.2**.**

[8, Theorem 1.3]* Let $E\subset\mathbb{Z}_{q}^{d}$, where $q=p^{l}$. Suppose*

[TABLE]

Then the distance set $\Delta(E)$ determined by the points of $E$ satisfies

[TABLE]

Later in [12], Hieu and Vinh proved the following distance result in the context of finite cyclic rings.

Theorem 1.3**.**

[12, Theorem 2.7]* Let $A\subset\mathbb{Z}_{q}$ be of cardinality $|A|\gg q^{1-\frac{1}{2r}}$. Then, the size of the distance set determined by $A^{n}$ satisfies*

[TABLE]

Now, let $R$ denote a finite valuation ring. In this paper, we study a variant of distance problem, namely pinned distance problem, for Cartesian product subsets $A\times A$ of $R^{2}$.

Note that, the method we use to prove the main result of this paper is analogous to the one given by Petridis in [16, Theorem 1.1]. More precisely, we first see pinned distances at a fixed point in $R^{2}$ as a point-plane incidence in $R^{3}$. Then we employ the point-plane incidence bound for multisets in $R^{3}$ which is recently given by Van The et al. in [17, Theorem 2.3]. This yields the lower bound for the size of the specified pinned distance set in $R^{2}$ in Theorem 1.4.

We should mention that the distance result we obtain in Theorem 1.4 recovers the pinned distance result in [16] in the finite filed setting. Also, in the setting of modules over finite cyclic rings $\mathbb{Z}_{q}$, it is an improvement on the distance results given in [8, 12] for the Cartesian product sets $A\times A\subset\mathbb{Z}_{q}^{2}.$

Before stating the main theorem, let us recall some necessary definitions.

1.1 Notation.

We note that a detailed definition of finite valuation ring can be found in [15]. In order to make the statements precise and self contained, we will review the definition and provide some key examples in this note. A finite valuation ring is a finite principal ideal domain which is local. Given a finite valuation ring $R$, we associate two parameters $q$ and $r$ to $R$ as follows. Let the maximal ideal $M$ of $R$ to be of the form $M=(\pi)$, where $\pi$ is the uniformizer of $R$, i.e. a non unit defined up to a unit of $R$. Let $r$ be the nilpotency degree of $\pi$, that is the smallest positive integer with the property that $\pi^{r}=0$ and $q$ be the size of the residue field $F=R/(\pi)$. Therefore, $R$ has the filtration

[TABLE]

where $|R|=q^{r}$. Some examples of finite valuation rings are as follows.

Finite fields $\mathbb{F}_{q}$, where $q=p^{k}$ is a prime power. 2. 2.

Finite cyclic rings $\mathbb{Z}_{p^{k}}$, where $p$ is a prime. 3. 3.

Function fields $F[x]/(f^{k})$, where $F$ is a finite field and $f$ is an irreducible polynomial in $F[x]$. 4. 4.

$\mathcal{O}/(p^{k})$, where $\mathcal{O}$ is the ring of integers in a number field and $p$ is a prime in $\mathcal{O}$.

Let us also write some of the examples above with parameters $q$ and $r$ as stated in the definition. Note that for the finite field $R=\mathbb{F}_{p^{k}}$, $p$ is a prime, we have $q=p^{k}$ and $r=1$. And for the finite cyclic ring $\mathbb{Z}_{p^{k}}$ we have the filtration

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Hence $r=k$ and $q=|\mathbb{Z}_{p^{k}}/(p)|=p$ in this case.

Next we recall the notion of distance in this context. For two points $u=(u_{1},\dots,u_{d})$ and $v=(v_{1},\dots v_{d})$ in $R^{d}$, the distance between them is given by

[TABLE]

For a subset $E\subset R^{d}$, the distance set determined by $E$ is

[TABLE]

and the distance set pinned at a fixed point $u$ of $E$ is defined by

[TABLE]

Throughout $R$ will denote a finite valuation of order $q^{r}$, where $q$ is an odd prime power. $X\gg Y$ means that there exists an absolute constant $c$ such that $X\geq cY$, and $"\ll"$ is defined similarly.

1.2 Statement of Main Result

Our main result is the following theorem.

Theorem 1.4**.**

Let $R$ be a finite valuation ring of order $q^{r}$, q is an odd prime power, and $A\subset R$. There exists a point $u\in A\times A\subset R^{2}$ such that

[TABLE]

In particular, if $|A|\geq q^{r-\frac{1}{3}}$, then $\Delta_{u}(A\times A)\gg q^{r}$ for some $u\in A\times A$ and hence $A\times A$ determines a positive proportion of all possible distances.

Remark 1.5*.*

Let $R=\mathbb{F}_{p}$, where $p$ is an odd prime. Note that in this case we can take $q=p$ and $r=1$ in Theorem 1.4 and conclude that if $A\subset\mathbb{F}_{p}$, then there exists $u\in A\times A\subset\mathbb{F}_{p}^{2}$ such that

[TABLE]

In particular, if $|A|>p^{\frac{2}{3}},$ then $|\Delta_{u}(A\times A)|\gg p$ for some $u\in A\times A$. This result matches with the result of Petridis given in [16, Theorem 1.1] in the context of prime fields and generalize it to the broader context of finite valuation rings.

Remark 1.6*.*

We note that the result in [8, Theorem 1.3] in the special case $d=2$ implies that if $E\subset\mathbb{Z}_{q}^{2}$, where $q=p^{l}$, and $|E|\gg l(l+1)q^{2-\frac{1}{2l}}$, then

[TABLE]

On the other hand, Theorem 1.4 implies that if $E=A\times A\subset\mathbb{Z}_{q}^{2}$, where $q=p^{l}$, and $|E|=|A\times A|\geq q^{2-\frac{2}{3l}}$, then $|\Delta(E)|=|\Delta(A\times A)|\gg q.$

Therefore, in terms of getting a positive proportion of all possible distances, Theorem 1.4 improves the result in [8, Theorem 1.3] for Cartesian product sets of the form $A\times A\subset\mathbb{Z}_{q}^{2}$.

Remark 1.7*.*

In [12, Theorem 2.7], for $n=2$, the following result was obtained for subsets of finite cyclic rings. Let $A\subset\mathbb{Z}_{q}$ be of cardinality $|A|\gtrsim q^{1-1/2r},$ where $q=p^{r}$. Then the number of distances determined by $A\times A$ satisfies

[TABLE]

Theorem 1.4 can be seen as a generalization of this result to finite valuation rings and a slight improvement in the context of finite cyclic rings.

2 Proof of Theorem 1.4

For the proof of Theorem 1.4, we will need the following lemma from [16]. We note here that though Petridis stated Lemma 2.1 for subsets of finite fields $\mathbb{F}_{q}$, it can be readily checked that the same proof applies for subsets of any finite valuation ring.

Lemma 2.1**.**

Let $E\subset R^{2}$ and $N$ be the number of solutions to

[TABLE]

where $u,v,w\in E$. Then there exists $u\in E$ such that $|\Delta_{u}(E)|\geq\frac{|E|^{3}}{N}.$

We will also use the following point-plane incidence bound in $R^{3}$ from [17].

Theorem 2.2**.**

[17, Theorem 2.3]* Let $Q,\Pi$ be weighted set of points and planes in $R^{3}$ with the weighted integer function $w$, both total weight $W$. Suppose that maximum weights are bounded by $w_{0}\geq 1$ Let the number of weighted incidences be*

[TABLE]

where

[TABLE]

Then the number $I_{w}$ of weighted incidences is bounded as follows:

[TABLE]

Proof of Theorem 1.4.

We first note that if we write $u=(u_{1},u_{2})$, $v=(v_{1},v_{2})$ and $w=(w_{1},w_{2})$, where $u_{i},v_{i},w_{i}\in A$, then the equation (2.1) can be written as

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which can be restated as

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Next we define a set of points $Q$ and a set of planes $\Pi$ in $R^{3}$ as follows:

[TABLE]

and

[TABLE]

Then it follows that the number of incidences $\left|I(Q,\Pi)\right|$ between $Q$ and $\Pi$ is equal to the number of solutions of the equation (2.2) which is $N$ in Lemma 2.1.

Note that the total weight of $Q$ and $\Pi$ are both $W=|A|^{3}$. Hence, Theorem 2.2 implies that

[TABLE]

Therefore, by Lemma 2.1, there exists $u\in A\times A$ such that

[TABLE]

which completes the proof of Theorem 1.4. ∎

Acknowledgments. The author would like to thank Brendan Murphy for valuable comments.