[Paper Review] Asymptotic upper bounds on progression-free sets in $\mathbb{Z}_p^n$

Dion Gijswijt|arXiv (Cornell University)|May 18, 2016

Limits and Structures in Graph Theory6 references3 citations

TL;DR

This paper establishes an asymptotic upper bound of $ O(p^{cn}) $, where $ c = 1 - rac{1}{18/log p} < 1 $, on the size of progression-free subsets in $ bZ_p^n $ for odd primes $ p $, using an optimized polynomial method. For $ p=3 $, this yields an upper bound of $ O(2.84^n) $ for affine caps in $ bF_3^n $, improving prior bounds and matching the best-known lower bounds in the exponent range.

ABSTRACT

We show that any subset of $\mathbb{Z}_p^n$ ($p$ an odd prime) without $3$-term arithmetic progression has size $O(p^{cn})$, where $c:=1-\frac{1}{18\log p}&lt;1$. In particular, we find an upper bound of $O(2.84^n)$ on the maximum size of an affine cap in $GF(3)^n$.

Motivation & Objective

• To establish tighter asymptotic upper bounds on the size of progression-free subsets in $ \bbZ_p^n $ for odd primes $ p $, improving upon earlier results.
• To extend the polynomial method of Croot, Lev, and Pach to $ \bbZ_p^n $, particularly for odd primes, beyond the $ \bbZ_4^n $ case.
• To provide a quantitative improvement over Meshulam’s $ O(p^n / n) $ bound and Bateman-Katz’s $ O(3^n / n^{1+\epsilon}) $ bound in the $ p=3 $ case.
• To derive a general upper bound of the form $ O(p^{cn}) $ with $ c < 1 $, showing exponential savings in the exponent compared to the trivial $ p^n $ bound.

Proposed method

• Uses the polynomial method with multivariate polynomials over $ \bbF_p $, restricting to monomials with individual variable degrees at most $ p-1 $.
• Applies Hoeffding’s inequality to estimate the dimension of the space of low-degree polynomials $ L_{n,d} $, specifically for $ d = \frac{1}{3}(p-1)n $.
• Introduces a symmetric bilinear form via the evaluation of $ f(x+y) $, leveraging the structure of $ A \times A $ matrices to bound progression-free sets.
• Constructs a subspace $ V = K \cap L $, where $ K $ vanishes on the sumset $ C = \{a+a \mid a \in A\} $, and $ L $ is the space of polynomials of degree at most $ \frac{2}{3}(p-1)n $, to control the size of $ A $.
• Uses rank bounds on the matrix $ M_{a,b} = f(a+b) $ to relate the size of $ A $ to the dimension of low-degree polynomial spaces via Proposition 2.
• Employs duality and dimension counting: $ \dim V \geq |C| - p^{cn} $, and shows $ |A| \leq 3p^{cn} $ by constructing a nonvanishing polynomial on $ C' \subseteq C $ of size $ \dim V $.

Experimental results

Research questions

• RQ1 What is the best possible asymptotic upper bound on the size of progression-free subsets in $ \bbZ_p^n $ for odd primes $ p $?
• RQ2 Can the polynomial method used for $ \bbZ_4^n $ be extended to $ \bbZ_p^n $ for odd $ p $, and what improvements does it yield?
• RQ3 How does the new bound compare to the previous best-known bounds of Meshulam and Bateman-Katz in the case $ p=3 $?
• RQ4 Can the exponent in the upper bound be improved below $ p^n $, and what is the optimal value of $ c $ such that $ |A| = O(p^{cn}) $?
• RQ5 What is the tightest known upper bound on the size of affine caps in $ \bbF_3^n $?

Key findings

• For any odd prime $ p $, the size of any progression-free subset in $ \bbZ_p^n $ is bounded by $ O(p^{cn}) $, where $ c = 1 - \frac{1}{18\log p} < 1 $, showing exponential savings over the trivial $ p^n $ bound.
• In the case $ p=3 $, the bound improves to $ O(2.84^n) $, which is tighter than the previous best-known bound of $ O(3^n / n^{1+\epsilon}) $ due to Bateman and Katz.
• The bound $ O(2.84^n) $ for affine caps in $ \bbF_3^n $ is within a constant factor of the best known lower bound of $ \Omega(2.2174^n) $, indicating strong asymptotic tightness.
• The method relies on a refined application of Hoeffding’s inequality to estimate the dimension of low-degree polynomial spaces, leading to the exponent $ c $.
• The proof establishes that the size of a progression-free set $ A $ satisfies $ |A| \leq 3p^{cn} $, with equality condition derived via dimension counting and rank arguments on the matrix $ f(a+b) $.
• The result was independently obtained by Jordan S. Ellenberg, and the two works were later merged into a joint publication, confirming the robustness and significance of the bound.