[Paper Review] Right Angles in (F_q)^d

Michael A. Bennett|arXiv (Cornell University)|Nov 28, 2015

Limits and Structures in Graph Theory9 references3 citations

TL;DR

This paper investigates Erdos-Falconer-type problems in finite vector spaces, proving that subsets of size >> q^{(d+2)/3} in (F_q)^d contain three points forming a right angle, and subsets of size >> q^{(d+2)/2} contain a right angle with vertex at the origin. The latter bound is shown to be sharp up to constants, with partial results on spread and collinear triples.

ABSTRACT

Here we examine some Erdos-Falconer-type problems in vector spaces over finite fields involving right angles. Our main goals are to show that a) a subset A of F_q^d of size >> q^[(d+2)/3] contains three points which generate a right angle, and b) a subset A of F_q^d of size >> q^[(d+2)/2] contains two points which generate a right angle with the vertex at the origin. We will also prove that b) is sharp up to constants and provide some partial results for similar problems related to spread and collinear triples.

Motivation & Objective

• To determine the minimal size of a subset A ⊆ (F_q)^d that guarantees the existence of three points forming a right angle.
• To establish the threshold size for a subset to contain a right angle with the vertex at the origin.
• To prove that the bound for the origin-vertex case is sharp up to constant factors.
• To explore related problems involving spread and collinear triples in finite fields.

Proposed method

• Utilizes additive combinatorics and Fourier analytic methods to analyze configurations in (F_q)^d.
• Applies the Cauchy-Schwarz inequality and exponential sum estimates to control the number of right-angled triples.
• Employs the method of moments and energy estimates to bound the number of solutions to angle-related equations.
• Leverages duality and symmetry in finite fields to reduce geometric problems to additive problems.
• Constructs extremal examples to demonstrate the sharpness of the q^{(d+2)/2} bound for origin-vertex right angles.
• Analyzes the geometry of quadratic forms over F_q to characterize right angles via dot product vanishing.

Experimental results

Research questions

• RQ1 What is the minimal size of a subset A ⊆ (F_q)^d that guarantees three points forming a right angle?
• RQ2 What is the threshold size for a subset to contain a right angle with the vertex at the origin?
• RQ3 Is the bound q^{(d+2)/2} for origin-vertex right angles sharp up to constant factors?
• RQ4 How do the results extend to configurations involving spread or collinear triples?
• RQ5 What geometric and algebraic structures in (F_q)^d govern the existence of right angles?

Key findings

• A subset A ⊆ (F_q)^d of size >> q^{(d+2)/3} must contain three points that form a right angle.
• A subset A ⊆ (F_q)^d of size >> q^{(d+2)/2} must contain a right angle with the vertex at the origin.
• The bound q^{(d+2)/2} for origin-vertex right angles is sharp up to constant factors, as demonstrated by extremal constructions.
• Partial results are obtained for problems related to spread and collinear triples, suggesting analogous thresholds may exist.
• The methods successfully transfer geometric problems in finite fields into additive combinatorics, enabling quantitative estimates.
• The use of exponential sums and Fourier analysis provides effective control over the number of right-angled configurations.