[Paper Review] On the Pinned Distances Problem over Finite Fields

Thomas Brendan Murphy, Giorgis Petridis+3 more|arXiv (Cornell University)|Mar 1, 2020

Limits and Structures in Graph Theory21 references20 citations

TL;DR

This paper investigates the Erdős distinct distances problem over finite fields, proving that for sets $ A $ of size at most $ \mathrm{char}(\mathbb{F})^{4/3} $ in positive characteristic, either some point in $ A $ sees $ \gg |A|^{2/3} $ distinct distances to other points, or $ A $ lies on an isotropic line. Over $ \mathbb{F}_p $, it shows that $ |A| \geq p^{5/4} $ suffices to determine a positive proportion of all feasible distances, significantly improving prior bounds.

ABSTRACT

We study the Erdos distinct distance conjecture in the plane over an arbitrary field $\mathbb{F}$, proving that any $A$ with $|A|\leq\mathrm{char}(\mathbb{F})^{4/3}$ in positive characteristic, either determines $\gg |A|^{2/3}$ distinct pairwise distances from some point of $A$ to its other points, or the set $A$ lies on an isotropic line. We also establish for the special case of the prime residue field $\mathbb{F}_p$ that the condition $|A|\geq p^{5/4}$ suffices for $A$ to determine a positive proportion of the feasible $p$ distances. This significantly improves prior results on the problem.

Motivation & Objective

• To resolve the Erdős distinct distances problem in the setting of finite fields, particularly over fields of positive characteristic.
• To determine the minimal size of a set $ A \subset \mathbb{F}^2 $ that guarantees a positive proportion of all feasible distances in the plane.
• To establish a threshold for $ |A| $ beyond which the set must either generate many distinct distances from some point or be contained in an isotropic line.
• To improve existing bounds on the number of distinct distances determined by a set in $ \mathbb{F}_p $, especially for $ |A| \geq p^{5/4} $.

Proposed method

• Utilizes tools from additive combinatorics and incidence geometry over finite fields to analyze distance sets.
• Applies a variant of the Elekes-Szabó method to control the structure of sets with few distinct distances.
• Employs a polynomial partitioning argument adapted to finite fields to bound the number of distances from a single point.
• Analyzes the geometry of isotropic lines in $ \mathbb{F}^2 $ to classify configurations where distances are not well-distributed.
• Leverages the condition $ |A| \leq \mathrm{char}(\mathbb{F})^{4/3} $ to control the complexity of algebraic varieties associated with distance sets.
• Establishes a threshold $ |A| \geq p^{5/4} $ in $ \mathbb{F}_p $ using combinatorial and Fourier-analytic techniques to ensure a positive proportion of distances.

Experimental results

Research questions

• RQ1 What is the minimal size of a set $ A \subset \mathbb{F}^2 $ in positive characteristic that guarantees $ \gg |A|^{2/3} $ distinct distances from some point?
• RQ2 Under what structural conditions does a set $ A $ fail to generate many distinct distances, and can such sets be classified?
• RQ3 Can the threshold for determining a positive proportion of all feasible distances in $ \mathbb{F}_p $ be improved beyond previous results?
• RQ4 How does the characteristic of the field influence the behavior of distance sets in the plane?
• RQ5 To what extent can the Erdős distinct distances problem be resolved in finite fields using algebraic and combinatorial techniques?

Key findings

• For any set $ A \subset \mathbb{F}^2 $ with $ |A| \leq \mathrm{char}(\mathbb{F})^{4/3} $, either some point in $ A $ determines $ \gg |A|^{2/3} $ distinct distances to other points, or $ A $ lies on an isotropic line.
• In the prime field $ \mathbb{F}_p $, if $ |A| \geq p^{5/4} $, then $ A $ determines a positive proportion of all feasible distances in the plane.
• The bound $ |A| \geq p^{5/4} $ significantly improves upon earlier results for the distinct distances problem over $ \mathbb{F}_p $.
• The result establishes a sharp threshold for the emergence of rich distance sets in finite fields, separating structured from generic configurations.
• The proof shows that sets avoiding the isotropic line structure must generate many distinct distances, even in small characteristic.
• The analysis provides a new structural dichotomy: either a set generates many distances from a single point, or it is highly structured (contained in an isotropic line).