The Constant of Proportionality in Lower Bound Constructions of Point-Line Incidences

TL;DR

This paper improves the lower bounds on the constant of proportionality in point-line incidence bounds, showing constructions that achieve constants of at least 1 and approximately 1.11, surpassing previous bounds.

Contribution

It introduces modified and new constructions that establish higher lower bounds for the constant of proportionality in incidence bounds.

Findings

Elekes' construction yields a lower bound of 0.63.

Modified Elekes' construction improves the bound to 1.

Erdős' construction further improves the bound to approximately 1.11.

Abstract

Let $I(n,l)$ denote the maximum possible number of incidences between $n$ points and $l$ lines. It is well known that $I(n,l) = \Theta(n^{2/3}l^{2/3} + n + l)$. Let $c_{\mathrm{SzTr}}$ denote the lower bound on the constant of proportionality of the $n^{2/3}l^{2/3}$ term. The known lower bound, due to Elekes, is $c_{\mathrm{SzTr}} \ge 2^{-2/3} = 0.63$. With a slight modification of Elekes' construction, we show that it can give a better lower bound of $c_{\mathrm{SzTr}} \ge 1$, i.e., $I(n,l) \ge n^{2/3}l^{2/3}$. Furthermore, we analyze a different construction given by Erd{\H o}s, and show its constant of proportionality to be even better, $c_{\mathrm{SzTr}} \ge 3/(2^{1/3}\pi^{2/3}) \approx 1.11$.

Equations

$$I(P,L)\le C_{\mathrm{SzTr}}{n}^{2/3}{l}^{2/3}.$$

$$I(P,L)\ge c_{\mathrm{SzTr}}{n}^{2/3}{l}^{2/3}.$$

$$\mathrm{Elekes}(k,m)=(P,L)$$

$$P=\left\{0,\dots ,k-1\right\}\times \left\{0,\dots ,km-1\right\},$$

$$0\le b\le km-1,$$

$$-\frac{b}{k-1}\le a\le m+\frac{m-1}{k-1}-\frac{b}{k-1}.$$

$$l\approx km\left(m+\lfloor \frac{m-1}{k-1}\rfloor +\frac{1+((m-1)\mathrm{mod}(k-1))}{k-1}\right),$$

$$l\ge k{m}^2.$$

$$n=k^{2}m.$$

$$k\ge \frac{n^{2/3}}{l^{1/3}}.$$

$$I=lk\ge n^{2/3}{l}^{2/3}$$

$$\mathrm{Erdos}(k,m)=(P,L)$$

$$P={\left\{0,\dots ,k-1\right\}}^2.$$

$$I\approx \frac{6k^2{m}^2}{{\pi }^2}.$$

$$l$$

$$\approx$$

$$l\approx \frac{4m^{3}k}{{\pi }^2}.$$

$$I\approx \frac{3}{2^{1/3}{\pi }^{2/3}}{n}^{2/3}{l}^{2/3}$$

Full text

1 Overview

Let $P$ be a set of $n$ points in ${\mathbb{R}}^{2}$, and let $L$ be a family of $l$ lines in ${\mathbb{R}}^{2}$. We denote the number of incidences between these points and lines by $I(P,L)$. We denote by $I(n,l)$ the maximum of $I(P,L)$ over all sets $P$ of $n$ points, and families $L$ of $l$ lines. The Szemerédi-Trotter bound [7] asserts that $I(n,l)=O(n^{2/3}l^{2/3}+n+l)$ (See also [1, 6] for simpler proofs). For values of $n$ and $l$ such that $\sqrt{n}\leq l\leq n^{2}$, the $n^{2/3}l^{2/3}$ term dominates, so the bound becomes $I(n,l)=O(n^{2/3}l^{2/3})$. In more detail, we have:

Theorem 1.1** (Szemerédi and Trotter [7]).**

There exists a constant $C_{\mathrm{SzTr}}$ such that, for any set $P$ of $n$ points, and any family $L$ of $l$ lines, if $\sqrt{n}\leq l\leq n^{2}$, then the number of incidences between the points and lines is at most

[TABLE]

The known upper bound on $C_{\mathrm{SzTr}}$ at present, due to Pach et al. [4], is $C_{\mathrm{SzTr}}\leq 2.5$. The bound of Theorem 1.1 is asymptotically tight, as shown in different lower bound constructions by Erdős [3] and Elekes [2]. We state this claim more formaly as follows.

Theorem 1.2** (Erdős [3], Elekes [2]).**

There exists a constant $c_{\mathrm{SzTr}}>0$, such that, for infinitely many values of $n$ and $l$, where $\sqrt{n}\leq l\leq n^{2}$, there exist pairs $(P,L)$, where $P$ is a set of $n$ points, and $L$ is a family of $l$ lines, such that the number of incidences between the points and lines is at least

[TABLE]

The known lower bound on $c_{\mathrm{SzTr}}$, due to Elekes [2], is $c_{\mathrm{SzTr}}\geq 2^{-2/3}=0.63$.

In this paper we improve the estimate of $c_{\mathrm{SzTr}}$. We modify Elekes’ construction, and show that this modification gives a lower of $c_{\mathrm{SzTr}}\geq 1$. Next, we analyze the construction of Erdős [3], and show its constant of proportionality to be even better, $c_{\mathrm{SzTr}}\geq 3/(2^{1/3}\pi^{2/3})\approx 1.11$. This is an improvement upon a previous analysis of the Erdős construction [5], which gives the bound $c_{\mathrm{SzTr}}\geq(3/(4\pi^{2}))^{1/3}\approx 0.42$.

2 The Elekes construction

Elekes [2] gave the following lower bound construction. Let $k$ and $m$ be some positive integers. Put $P=\left\{1,\ldots,k\right\}\times\left\{1,\ldots,2km\right\}$, and put $L$ to be all lines $y=ax+b$, where $a\in\left\{1,\ldots,m\right\}$, and $b\in\left\{1,\ldots,km\right\}$. There are $n=|P|=2k^{2}m$ points and $l=|L|=km^{2}$ lines here, and each line is incident to exactly $k$ points, so $I=I(P,L)=k^{2}m^{2}$. It is then easy to verify that $I=2^{-2/3}n^{2/3}l^{2/3}$, and also, whenever $m>1$, that $\sqrt{n}\leq l\leq n^{2}$. This gives a lower bound on the $c_{\mathrm{SzTr}}$ constant from Theorem 1.2 of $c_{\mathrm{SzTr}}\geq 2^{-2/3}\approx 0.63$.

We present a slightly different construction from the above. It is similar in principle, but more exhaustive.

Definition 2.1**.**

Let $k$ and $m$ be some positive integers. We denote by

[TABLE]

the following set of points $P$, and family of lines $L$. $P$ is defined as a $k\times km$ lattice section:

[TABLE]

and $L$ is defined as all $x$-monotone lines that contain $k$ points of $P$.

With this definition of $\mathrm{Elekes}(k,m)$, we have $I(P,L)\geq|P|^{2/3}|L|^{2/3}$, and hence, $c_{\mathrm{SzTr}}\geq 1$. More formally:

Theorem 2.2**.**

Let $P$ and $L$ respectively be the points and lines of an $\mathrm{Elekes}(k,m)$ configuration, for some positive integers $k>1$ and $m$. Let us denote the number of points by $|P|=n$, the number of lines by $|L|=l$, and the number of incidences between them by $I(P,L)=I$. Then $I\geq n^{2/3}l^{2/3}$.

Proof.

The lines of $L$ have the form $y=ax+b$ with integer parameters as follows. The $b$ parameter is an integer in the range

[TABLE]

and the $a$ parameter, given $b$, is restricted as follows. For $x=k-1$ we have $0\leq a(k-1)+b\leq km-1$, or

[TABLE]

The difference between the upper and lower bounds of $a$ is $m+(m-1)/(k-1)$, and the number of integer values in this range is either $m+\left\lfloor(m-1)/(k-1)\right\rfloor$, or $m+1+\left\lfloor(m-1)/(k-1)\right\rfloor$. The latter case happens about $1+((m-1)\mod(k-1))$ out of $k-1$ times. The number of lines, resulting from multimplying the number of $b$-values by the number of $a$-values, is

[TABLE]

and in any event it is greater than $km^{2}$,

[TABLE]

The number of points is

[TABLE]

It then follows that

[TABLE]

Since each line is incident to $k$ points, the number of incidences comes out

[TABLE]

as claimed. This completes the proof. ∎

From this theorem it follows that $c_{\mathrm{SzTr}}\geq 1$. Note that an $\mathrm{Elekes}(k,k-1)$ has an equal number of points and lines, $n=l=k^{2}(k-1)$, and $I=k^{3}(k-1)\approx n^{4/3}$ incidences.

3 The Erdős construction

Erdős [3] considered $n$ points on a $n^{1/2}\times n^{1/2}$ lattice section, together with the $n$ lines that contain the most points. He noted that there are $\Theta(n^{4/3})$ incidences in this configuration, and conjectured that it is asymptotically optimal. His conjecture was settled in the affirmative as a corollary of the Szemerédi-Trotter bound [7]. Pach and Tóth [5] analyzed, in more generality, the square lattice section together with the lines with the most incidences, where the number of lines $l$ is not necessarily equal to the number of points $n$. Their analysis yielded the bound $I\geq 0.42n^{2/3}l^{2/3}$. In this section we will analyze the same setting in a different way and get an improved bound of $I\geq 1.11n^{2/3}l^{2/3}$, i.e., $c_{\mathrm{SzTr}}\geq 1.11$.

First, we give a formal definition of the Erdős construction.

Definition 3.1**.**

For two positive integers $k$ and $m$, we denote by

[TABLE]

the following set of points $P$, and family of lines $L$. We put $P$ to be a $k\times k$ lattice section:

[TABLE]

Next, we put $L$ to be all lines of the form $ax+by=c$ that pass through the bounding square of $P$, where:

$a$, $b$, and $c$ are integers. 2. 2.

$a$* and $b$ are coprime.* 3. 3.

$a\geq 0$. 4. 4.

$|a|+|b|\leq m$.

Under this definition, $L$ is not quite the family of lines with the most incidences with respect to $P$, but rather, an approximation of it. Indeed, there are lines here, such as $x+y=0$, with just one incidence. There are even lines with no incidences, like $2x+3y=1$ (this line exists whenever $k\geq 2$, and $m\geq 5$). However, most lines do have many incidences, which gives us the following result.

Theorem 3.2**.**

Let $P$ and $L$ respectively be the points and lines of an $\mathrm{Erdos}(k,m)$ configuration, for some positive integers $k$ and $m$. Let us denote the number of points by $|P|=n$, the number of lines by $|L|=l$, and the number of incidences between them by $I(P,L)=I$. Then $I\approx\frac{3}{2^{1/3}\pi^{2/3}}n^{2/3}l^{2/3}$.

The notation $\Phi\approx\Psi$, where both expressions depend on some set of variable $x_{1},x_{2},\ldots$ is shorthand for $\lim_{x_{1}\to\infty,x_{2}\to\infty,\ldots}(\Phi/\Psi)=1$. That is, as the independent variables (in the case of Theorem 3.2, $k$ and $m$) grow larger and larger, the ratio between the two expressions ($I$ and $\frac{3}{2^{1/3}\pi^{2/3}}n^{2/3}l^{2/3}$, in the case of Theorem 3.2) gets closer and closer to one.

Proof.

The number of points is $n=k^{2}$. The probability of a random pair $(a,b)$ to be coprime is about $\frac{6}{\pi^{2}}$ [8]. There are $(m+1)^{2}$ integer pairs in the range $\left\{\left.(a,b)\;\right||a|+|b|\leq m,a\geq 0\right\}$, so there are about $\frac{6m^{2}}{\pi^{2}}$ coprime pairs. Each pair $(a,b)$ determines the direction of a pencil of parallel lines, $ax+by=c$, and each of the $k^{2}$ points is incident to a line in each of these directions. That is, each point is incident to about $\frac{6m^{2}}{\pi^{2}}$ lines, so in total

[TABLE]

It remains to estimate the number of lines. Consider a positive coprime pair $(a,b)$. This pair generates lines $ax+by=c$, where:

The minimal value of $c$ is [math], and the line $ax+by=0$ passes through $(0,0)\in P$. 2. 2.

The maximal value of $c$ is $(a+b)(k-1)$, and the line $ax+by=(a+b)(k-1)$ passes through $(k-1,k-1)\in P$.

It follows that there are $(|a|+|b|)(k-1)+1$ values of $c$ that generate lines that pass through the square. This number of lines is true also for negative $b$ with a different range of $c$-values. The total number of lines $|L|=l$ is thus

[TABLE]

(3.1) is a sum over all coprime pairs $(a,b)$ as above. (3.2) is the same sum in a different order of summation. In (3.3) we estimate the number of coprime pairs $(a,b)$ such that $|a|+|b|=j$ as follows. There are $2j+1$ integer pairs $(a,b)$, such that $a\geq 0$ and $|a|+|b|=j$, and the probability of a pair from this subset to be coprime is, as already noted, $6/\pi^{2}$, so there should be an expected number of $(12j+6)/\pi^{2}\approx 12j/\pi^{2}$ coprime pairs. In (3.5) we use the approximation $\sum_{j=1}^{m}j^{2}=m(m+1)(2m+1)/6\approx m^{3}/3$. The dominant term in the final equation is

[TABLE]

From the values of $n,l$, and $I$ in terms of $k$ and $m$, we get that

[TABLE]

as claimed. This copmletes the proof. ∎

From Theorem 3.2 it follows that $c_{\mathrm{SzTr}}\geq\frac{3}{2^{1/3}\pi^{2/3}}\approx 1.11$.