On diameter bounds for planar integral point sets in semi-general position

TL;DR

This paper establishes a new lower bound, better than linear, for the minimum diameter of planar integral point sets in semi-general position, advancing understanding of their geometric constraints.

Contribution

It introduces a novel lower bound for the diameter of semi-general position point sets, improving upon previous linear bounds.

Findings

New lower bound for diameter surpasses linear growth

Improved understanding of geometric constraints in integral point sets

Advances theoretical bounds in Euclidean geometry

Abstract

A point set $M$ in the Euclidean plane is called a planar integral point set if all the distances between the elements of $M$ are integers, and $M$ is not situated on a straight line. A planar integral point set is called to be in semi-general position, if it does not contain collinear triples. The existing lower bound for mininum diameter of planar integral point sets is linear. We prove a new lower bound for mininum diameter of planar integral point sets in semi-general position that is better than linear.

Equations

$$\mathrm{diam}M=\underset{A,B\in M}{\max }|AB|,$$

$$d(2,n)=\underset{M\in \mathfrak{M}(2,n)}{\min }\mathrm{diam}M.$$

$$d(2,n)\le \overline{d}(2,n)\le \dot{d}(2,n),$$

$$c_{1}n\le d(2,n)\le \overline{d}(2,n)\le n^{c_2\log \log n}.$$

$$\mathrm{diam}P\ge n^{\frac{\delta }{4\log 2(1+\epsilon )}\log \log n}.$$

$$\begin{gathered} 4(|M_1{M}_2|-1)(|M_3{M}_4|-1)+4+4=4((|M_1{M}_2|-1)(|M_3{M}_4|-1)+2)= \\ =4(|M_1{M}_2|\cdot |M_3{M}_4|+1-|M_1{M}_2|-|M_3{M}_4|+2)= \\ =4(|M_1{M}_2|\cdot |M_3{M}_4|+(1-|M_1{M}_2|)+(2-|M_3{M}_4|))\le 4|M_1{M}_2|\cdot |M_3{M}_4|, \end{gathered}$$

$$\overline{d}(2,n)\ge (n/5{)}^{5/4}.$$

$$\underset{A,B\in M}{\min }|AB|=|M_1{M}_2|,$$

$$\underset{A,B\in M\setminus \{M_1\}}{\min }|AB|=|M_3{M}_4|=m.$$

$$n\le 4\cdot |M_1{M}_2|\cdot |M_3{M}_4|\le 4p^{4/5},$$

$$p\ge (n/4{)}^{5/4}>(n/5{)}^{5/4}.$$

$$n\le 2(2p^{4/5}+1)+1=4p^{4/5}+3\le 5p^{4/5}.$$

$$p\ge (n/5{)}^{5/4}.$$

$$n\le 3\frac{\mathrm{diam}H}{n^{1/6}},$$

$$\overline{d}(2,n)\ge c_3{n}^{7/6},$$

Full text

On diameter bounds for planar integral point sets in semi-general position

111 This work was carried out at Voronezh State University and supported by the Russian Science Foundation grant 19-11-00197.

N.N. Avdeev 222nickkolok@mail.ru, avdeev@math.vsu.ru

Abstract.

A point set $M$ in the Euclidean plane is called a planar integral point set if all the distances between the elements of $M$ are integers, and $M$ is not situated on a straight line. A planar integral point set is called to be in semi-general position, if it does not contain collinear triples. The existing lower bound for mininum diameter of planar integral point sets is linear. We prove a new lower bound for mininum diameter of planar integral point sets in semi-general position that is better than linear.

1 Introduction

An integral point set in a plane is a point set $M$ such that all the usual (Euclidean) distances between the points of $M$ are integers and $M$ is not situated on a straight line. Every integral point set consists of a finite number of points [1, 2]; thus, we denote the set of all planar integral point sets of $n$ points by $\mathfrak{M}(2,n)$ (using the notation in [3]) and define the diameter of $M\in\mathfrak{M}(2,n)$ in the following natural way:

[TABLE]

where $|AB|$ denotes the Euclidean distance. The symbol $\#M$ will be used for cardinality of $M$, that is the number of points in $M$ in our case.

Since every integral point set can obviously be dilated to a set of larger diameter, minimal possible diameters of sets of given cardinality are in the focus. To be precise, the following function was introduced [4, 5]:

[TABLE]

It turned out to be very easy to construct a planar integral point set of $n$ points with $n-1$ collinear ones and one point out of the line (so-called facher sets); the same holds for 2 points out of the line (we refer the reader to [6], where some of such sets are called crabs) and even for 4 points out of the line [7]. For $9\leq n\leq 122$, the minimal possible diameter is achieved at a facher set [4].

Definition 1.1**.**

A set $M\in\mathfrak{M}(2,n)$ is called to be in semi-general position, if no three points of $M$ are collinear. The set of all planar integral point sets in semi-general position is denoted by $\overline{\mathfrak{M}}(2,n)$.

Furthermore, the constructions of integral point sets in semi-general position of arbitrary cardinality appeared [8]; such sets are situated on a circle. Also, there is a sophisticated construction of a circular integral point set of arbitrary cardinality that gives the possible numbers of odd integral distances between points in the plane [9].

Definition 1.2**.**

A set $M\in\overline{\mathfrak{M}}(2,n)$ is called to be in general position, if no four points of $M$ are concyclic. The set of all planar integral point sets in general position is denoted by $\dot{\mathfrak{M}}(2,n)$.

It remains unknown if there are integral points sets in general position of arbitrary cardinality; however, some sets $M\in\dot{\mathfrak{M}}(2,7)$ are known [10, 11].

The inequality

[TABLE]

where $\overline{d}(2,n)=\min_{M\in\overline{\mathfrak{M}}(2,n)}\operatorname{diam}M$ and $\dot{d}(2,n)=\min_{M\in\dot{\mathfrak{M}}(2,n)}\operatorname{diam}M$, is obvious; however, a more interesting relation holds:

[TABLE]

The upper bound is presented in [8]. The lower bound was firstly introduced in [12]; the largest known value for $c_{1}$ is $5/11$ for $n\geq 4$ [13].

There are some bounds for minimal diameter of planar integral point sets in some special positions. Assuming that the planar integral point sets contains many collinear points, the following result holds.

Theorem 1.3**.**

[5, Theorem 4]** For $\delta>0$, $\varepsilon>0$, and $P\in\mathfrak{M}(2,n)$ with at least $n^{\delta}$ collinear points, there exists a $n_{0}(\varepsilon)$ such that for all $n\geq n_{0}(\varepsilon)$ we have

[TABLE]

For diameter bounds for circular sets, we refer the reader to [14].

Particular cases of planar integral point sets are also discussed in [15, §5.11], [16, §D20], [17], [18]. For generalizaton in higher dimensions and the appropriate bounds, see [19, 20].

In the present paper we give a special bound for planar integral point sets in semi-general position. The condition of semi-general position is important in the given proof.

2 Preliminary results

In this section, we give some lemmas which will be used for the proof.

Lemma 2.1**.**

[12, Observation 1]** If a triangle $T$ has integer side-lengths $a\leq b\leq c$, then the minimal height $m$ of it is at least $\left(a-\frac{1}{4}\right)^{1/2}$.

Definition 2.2**.**

The part of a plane between two parallel straight lines with distance $\rho$ between the lines is called a strip of width $\rho$.

Lemma 2.3**.**

[21]** If a triange $T$ with minimal height $\rho$ is situated in a strip, then the width of a strip is at least $\rho$.

Corollary 2.4**.**

If a triangle $T$ with integer side-lengths $a\leq b\leq c$ is situated in a strip, then the width of a strip is at least $\left(a-\frac{1}{4}\right)^{1/2}$.

Lemma 2.5**.**

[3*, Lemma 4]**; [13, Lemma 2.4]

Let $M\in\mathfrak{M}(2,n)$, $\operatorname{diam}M=d$. Then $M$ is situated in a square of side length $d$.*

Definition 2.6**.**

[13, Definition 2.5] A cross for points $M_{1}$ and $M_{2}$, denoted by $cr(M_{1},M_{2})$, is the union of two straight lines: the line through $M_{1}$ and $M_{2}$, and the perpendicular bisector of line segment $M_{1}M_{2}$.

Lemma 2.7**.**

[13*, Theorem 3.10]**

Each set $M\in\mathfrak{M}(2,n)$ such that for some $M_{1},M_{2}\in M$ equality $|M_{1}M_{2}|=1$ holds, consists of $n-1$ points, including $M_{1}$ and $M_{2}$, on a straight line, and one point out of the line, on the perpendicular bisector of line segment $M_{1}M_{2}$.*

Lemma 2.8**.**

Let $\{M_{1},M_{2},M_{3},M_{4}\}\subset M\in\overline{\mathfrak{M}}(2,n)$ (points $M_{2}$ and $M_{3}$ may coincide, other points may not), $n\geq 4$. Then $\#M\leq 4\cdot|M_{1}M_{2}|\cdot|M_{3}M_{4}|$.

Remark 2.9**.**

Lemma 2.8 is one of the variations of [2].

Proof.

Each point $N\in M$ satisfies one of the following conditions:

a) $N$ belongs to $cr(M_{1},M_{2})$ — overall at most 4 points;

b) $N$ belongs to $cr(M_{3},M_{4})$ — overall at most 4 points;

c) $N$ belongs to the intersection of one of $|M_{1}M_{2}|-1$ hyperbolas with one of $|M_{3}M_{4}|-1$ hyperbolas — overall at most $4(|M_{1}M_{2}|-1)(|M_{3}M_{4}|-1)$ points;

Due to Lemma 2.7 we have $|M_{1}M_{2}|\geq 2$ and $|M_{3}M_{4}|\geq 2$. Since

[TABLE]

we are done. ∎

3 The main result

Theorem 3.1**.**

For every integer $n\geq 3$ we have

[TABLE]

Proof.

For $n=3$ we have $\overline{d}(2,3)=1$. Consider $M\in\overline{\mathfrak{M}}(2,n)$, $n\geq 4$, $\operatorname{diam}M=p$.

Let us choose points $M_{1},M_{2},M_{3},M_{4}\in M$ (points $M_{2}$ and $M_{3}$ may coincide, other points may not), such that

[TABLE]

[TABLE]

For $m\leq p^{2/5}$, Lemma 2.8 yields that

[TABLE]

or, that is the same,

[TABLE]

So, let us consider $m>p^{2/5}$. Then for any $A,B\in M\setminus\{M_{1}\}$ the inequality $|AB|>p^{2/5}$ holds. Due to Corollary 2.4, no three points of $M\setminus\{M_{1}\}$ are located in a strip of width $p^{1/5}/2$.

Lemma 2.5 yields that $M$ is situated in a square with side length $p$. Let us partition this square into $q$ strips, $2p^{4/5}\leq q<2p^{4/5}+1$, each of width at most $p^{1/5}/2$. Every strip contains at most two points of $M\setminus\{M_{1}\}$, thus

[TABLE]

The latter inequality holds because $\overline{d}(2,n)\geq 4$ for $n\geq 4$  [5] and $4^{4/5}>3$. From the inequality (10) one can easily derive that

[TABLE]

∎

Remark 3.2**.**

The following result in known:

Lemma 3.3**.**

[12, Corollary 1]** For $H\in\overline{\mathfrak{M}}(2,n)$, the minimum distance in H is at least $n^{1/3}$.

Applying the same technique, one can easily derive that

[TABLE]

which leads to the bound

[TABLE]

which is less than the one from Theorem 3.1.

4 Conclusion

The presented bound is the first special lower bound for sets in semi-general position. Thus, we did not accepted the challenge to make the constant in Theorem 3.1 as large as possible, in order to keep the ideas of the proof clear and understandable. A more thorough research can be done in the future to enlarge the constant. However, the upper and lower bounds are still not tight.

5 Acknowledgements

Author thanks Dr. Prof. E.M. Semenov for proofreading and valuable advice.