[Paper Review] On sets defining few ordinary lines

Ben Green, Terence Tao|arXiv (Cornell University)|Aug 23, 2012

Computational Geometry and Mesh Generation18 references4 citations

TL;DR

This paper resolves the Dirac-Motzkin conjecture by proving that any set of $ n $ points in the plane, not all collinear, spans at least $ n/2 $ ordinary lines (lines through exactly two points) for sufficiently large $ n $. It establishes a structure theorem showing that sets with few ordinary lines must be almost entirely contained in a cubic curve, and uses this to also solve the orchard-planting problem, proving the maximum number of 3-rich lines is $ \lfloor n(n-3)/6 \rfloor + 1 $ for large $ n $. The results are sharp and classify all extremal configurations.

ABSTRACT

Let P be a set of n points in the plane, not all on a line. We show that if n is large then there are at least n/2 ordinary lines, that is to say lines passing through exactly two points of P. This confirms, for large n, a conjecture of Dirac and Motzkin. In fact we describe the exact extremisers for this problem, as well as all sets having fewer than n - C ordinary lines for some absolute constant C. We also solve, for large n, the "orchard-planting problem", which asks for the maximum number of lines through exactly 3 points of P. Underlying these results is a structure theorem which states that if P has at most Kn ordinary lines then all but O(K) points of P lie on a cubic curve, if n is sufficiently large depending on K.

Motivation & Objective

• To resolve the long-standing Dirac-Motzkin conjecture on the minimum number of ordinary lines determined by $ n $ non-collinear points in the plane.
• To classify all configurations of $ n $ points that achieve fewer than $ n - C $ ordinary lines for some absolute constant $ C $.
• To solve the orchard-planting problem by determining the maximum number of 3-rich lines (lines through exactly three points) for large $ n $.
• To establish a structure theorem showing that sets with $ O(n) $ ordinary lines are almost entirely contained in a cubic curve.

Proposed method

• Uses projective duality and Euler's formula, inspired by Melchior’s proof of the Sylvester-Gallai theorem, to analyze line configurations.
• Applies a structure theorem stating that if a point set has at most $ Kn $ ordinary lines, then all but $ O(K) $ points lie on a cubic curve for large $ n $.
• Employs tools from additive combinatorics and root-of-unity geometry to bound the number of lines through a fixed point and pairs of points on regular polygons.
• Uses pigeonholing and trigonometric bounds on linear forms in angles to show that no point (except origin) lies on more than $ O(n^{5/6}) $ lines joining vertices of a regular $ n $-gon.
• Leverages the classification of zero-sum tuples of 12th roots of unity to control configurations with many concurrencies.
• Combines double-counting identities between ordinary lines and 3-rich lines to convert lower bounds on $ N_2 $ into upper bounds on $ N_3 $.

Experimental results

Research questions

• RQ1 What is the minimum number of ordinary lines that a set of $ n $ non-collinear points in the plane can determine, for large $ n $?
• RQ2 What configurations achieve this minimum, and what is the exact structure of such extremal sets?
• RQ3 What is the maximum number of 3-rich lines (lines through exactly three points) that a set of $ n $ points can have, for large $ n $?
• RQ4 How do configurations with few ordinary lines relate to algebraic curves, particularly cubic curves?
• RQ5 Can the structure of point sets with $ O(n) $ ordinary lines be fully characterized?

Key findings

• For sufficiently large $ n $, any set of $ n $ non-collinear points in the plane spans at least $ n/2 $ ordinary lines, confirming the Dirac-Motzkin conjecture.
• When $ n $ is odd, the minimum number of ordinary lines is at least $ 3\lfloor n/4 \rfloor $, which is strictly greater than $ n/2 $.
• The extremal configurations achieving $ n/2 $ ordinary lines for even $ n $ consist of $ n/2 $ points on a circle and $ n/2 $ points at infinity, forming a symmetric configuration.
• All sets with fewer than $ n - C $ ordinary lines for some absolute constant $ C $ are almost entirely contained in a cubic curve, with only $ O(K) $ points outside for $ Kn $ ordinary lines.
• The maximum number of 3-rich lines in a set of $ n $ points is $ \lfloor n(n-3)/6 \rfloor + 1 $ for large $ n $, and this bound is tight.
• The only extremal examples achieving equality in the orchard problem for large $ n $ are derived from irreducible cubic curves, such as the Fermat cubic or elliptic curves with rational points.