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Sets preserved by a large subgroup of the special linear group

Le Quang Hung, Thang Pham, Kaloyan Slavov

TL;DR

The paper studies subsets $E$ of the affine plane ${\mathbb F}_q^2$ that are highly stable under many elements of $\mathrm{SL}_2({\mathbb F}_q)$ by analyzing the stabilizer set $R_E$. It proves a sharp threshold: if $|E|\le c_1 q^{\alpha}$ and $|R_E|\ge c_2 q^{\beta}$ with $\beta\ge 3\alpha/2$, then $E$ must lie on a line, with sharpness examples showing the bound is tight in general. The proof combines a reduction to a point-line incidence problem in ${\mathbb F}_q^3$ via a Mockenhaupt--Tao incidence bound and, in the prime-field case, a group-theoretic orbit-stabilizer argument to bound $|R_E|$; together these yield a strong rigidity phenomenon for extremal configurations. The results illuminate a discrete rigidity analogue to packing-type phenomena and open avenues for higher-dimensional extensions and real-field analogues.

Abstract

Let $E$ be a subset of the affine plane over a finite field $\mathbb{F}_q$. We bound the size of the subgroup of $SL_2(\mathbb{F}_q)$ that preserves $E$. As a consequence, we show that if $E$ has size $\ll q^α$ and is preserved by $\gg q^β$ elements of $SL_2(\mathbb{F}_q)$ with $β\geq 3α/2$, then $E$ is contained in a line. This result is sharp in general, and will be proved by using combinatorial arguments and applying a point-line incidence bound in $\mathbb{F}_q^3$ due to Mockenhaupt and Tao (2004).

Sets preserved by a large subgroup of the special linear group

TL;DR

The paper studies subsets of the affine plane that are highly stable under many elements of by analyzing the stabilizer set . It proves a sharp threshold: if and with , then must lie on a line, with sharpness examples showing the bound is tight in general. The proof combines a reduction to a point-line incidence problem in via a Mockenhaupt--Tao incidence bound and, in the prime-field case, a group-theoretic orbit-stabilizer argument to bound ; together these yield a strong rigidity phenomenon for extremal configurations. The results illuminate a discrete rigidity analogue to packing-type phenomena and open avenues for higher-dimensional extensions and real-field analogues.

Abstract

Let be a subset of the affine plane over a finite field . We bound the size of the subgroup of that preserves . As a consequence, we show that if has size and is preserved by elements of with , then is contained in a line. This result is sharp in general, and will be proved by using combinatorial arguments and applying a point-line incidence bound in due to Mockenhaupt and Tao (2004).
Paper Structure (5 sections, 12 theorems, 26 equations)

This paper contains 5 sections, 12 theorems, 26 equations.

Table of Contents

  1. Introduction
  2. Proof of Theorem \ref{['main-thm']}
  3. Proof of Proposition \ref{['Prop:gp_theory']}
  4. Some discussions
  5. Acknowledgements

Key Result

Theorem 1.3

Given $c_1>0$, there exists a constant $c_2>0$ with the following property. Let $\alpha,\beta\geq 0$ be such that $\beta\geq 3\alpha/2$. Let $E$ be a subset of ${\mathbb F}_q^2$ with $|E|\leq c_1 q^{\alpha}$. Suppose the set $R_E$ of matrices $\theta$ in $\mathop{\mathrm{SL}}\nolimits_2(\mathbb{F}_q

Theorems & Definitions (26)

  • Example 1
  • Example 2
  • Example 3
  • Remark 1.2
  • Theorem 1.3
  • Corollary 1.4
  • Theorem 1.5
  • Example 4
  • Example 5
  • Proposition 1.6
  • ...and 16 more