[Paper Review] On large subsets of $F_q^n$ with no three-term arithmetic progression

Jordan S. Ellenberg, Dion Gijswijt|arXiv (Cornell University)|May 30, 2016

Limits and Structures in Graph Theory5 references18 citations

TL;DR

This paper applies the polynomial method—originally developed by Croot, Lev, and Pach—to prove that the size of the largest subset of $\mathbb{F}_q^n$ with no three-term arithmetic progression is bounded by $c^n$ for some $c < q$, resolving a long-standing open problem in additive combinatorics. For $q=3$, it establishes an exponential bound of $o(2.756^n)$, significantly improving prior results and settling the cap problem in the negative for exponential growth above $2.756^n$.

ABSTRACT

In this note, we show that the method of Croot, Lev, and Pach can be used to bound the size of a subset of $F_q^n$ with no three terms in arithmetic progression by $c^n$ with $c &lt; q$. For $q=3$, the problem of finding the largest subset with no three terms in arithmetic progression is called the `cap problem'. Previously the best known upper bound for the cap problem, due to Bateman and Katz, was $O(3^n / n^{1+ε})$.

Motivation & Objective

• To resolve the cap problem in $\mathbb{F}_3^n$ by establishing an upper bound on the size of subsets with no three-term arithmetic progression.
• To extend the Croot-Lev-Pach polynomial method to general finite fields $\mathbb{F}_q^n$ for $q$ odd.
• To prove that $r_3((\mathbb{Z}/p\mathbb{Z})^n)^{1/n}$ is bounded away from $p$ as $n \to \infty$, showing no exponential growth near $p^n$.
• To close the gap between the best known upper and lower bounds for cap sets in $\mathbb{F}_3^n$, particularly by improving on the $O(3^n/n^{1+\epsilon})$ bound.

Proposed method

• Adapts the Croot-Lev-Pach polynomial method to $\mathbb{F}_q^n$ by defining a space $S_n^d$ of polynomials of degree at most $d$ with individual variable degrees $\leq q-1$.
• Uses a generalized version of the key lemma from Croot, Lev, and Pach to bound the number of points where a polynomial $P$ of degree $\leq d$ is nonzero on a set $A$, under the condition that $P(\alpha a + \beta b) = 0$ for distinct $a,b \in A$.
• Applies a rank argument to a matrix $B_{ab} = P(\alpha a + \beta b)$, showing its rank is at most $2m_{d/2}$, where $m_{d/2}$ is the dimension of the space of degree-$d/2$ monomials.
• Constructs a polynomial $P$ vanishing on the complement of $-\gamma A$, and uses the support bound from Proposition 2 to derive the inequality $|A| \leq 2m_{d/2} + (q^n - m_d)$.
• Optimizes the bound by setting $d = 2(q-1)n/3$, leading to $|A| \leq 3m_{(q-1)n/3}$, and uses large deviation theory to show $m_{(q-1)n/3}/q^n$ decays exponentially.
• Applies Cramér's theorem to compute the rate function $I((q-1)/3)$, proving $m_{(q-1)n/3} = O(c^n)$ for $c < q$, yielding the final exponential bound.

Experimental results

Research questions

• RQ1 Can the Croot-Lev-Pach polynomial method be extended from $\mathbb{F}_4^n$ to general $\mathbb{F}_q^n$ for odd $q$?
• RQ2 Is the size of the largest cap set in $\mathbb{F}_3^n$ bounded by $c^n$ for some $c < 3$?
• RQ3 Does $r_3((\mathbb{Z}/p\mathbb{Z})^n)^{1/n}$ remain bounded away from $p$ as $n$ grows, ruling out exponential growth near $p^n$?
• RQ4 Can the upper bound for cap sets be improved beyond the $O(3^n/n^{1+\epsilon})$ result of Bateman and Katz?
• RQ5 What is the best possible exponential base $c$ such that $|A| = O(c^n)$ for subsets $A \subset \mathbb{F}_3^n$ with no three-term arithmetic progression?

Key findings

• The paper proves that for any odd prime power $q$, the size of the largest subset of $\mathbb{F}_q^n$ with no three-term arithmetic progression is $O(c^n)$ for some $c < q$, resolving the growth rate question in the negative for exponential growth near $q^n$.
• For $q = 3$, the bound is quantified as $|A| = o(2.756^n)$, significantly improving the previous best upper bound of $O(3^n/n^{1+\epsilon})$.
• The method establishes that $m_{(q-1)n/3}/q^n$ decays exponentially, with the decay rate determined by the rate function $I((q-1)/3)$ from large deviation theory.
• The key inequality $|A| \leq 3m_{(q-1)n/3}$ is derived via a polynomial rank argument and support counting, forming the core of the exponential bound.
• The result shows that $r_3((\mathbb{Z}/p\mathbb{Z})^n)^{1/n}$ is bounded away from $p$ as $n \to \infty$, implying no cap sets of size $\sim p^n$ exist.
• The bound $c < 2.756$ for $q=3$ is computed explicitly using the solution to the optimization problem in Cramér's theorem, with $e^\theta = (\sqrt{33} - 1)/8$.