New lower bounds for three-term progression free sets in $\mathbb{F}_p^n$

Abstract

We prove new lower bounds on the maximum size of sets $A\subseteq \mathbb{F}_p^n$ or $A\subseteq \mathbb{Z}_m^n$ not containing three-term arithmetic progressions (consisting of three distinct points). More specifically, we prove that for any fixed integer $m\ge 2$ and sufficiently large $n$ (in terms of $m$), there exists a three-term progression free subset $A\subseteq \mathbb{Z}_m^n$ of size $|A|\ge (cm)^n$ for some absolute constant $c>1/2$. Such a bound for $c=1/2$ can be obtained with a classical construction of Salem and Spencer from 1942, and improving upon this value of $1/2$ has been a well-known open problem (our proof gives $c= 0.54$). Our construction relies on finding a subset $S\subset \mathbb{Z}_m^2$ of size at least $(7/24)m^2$ with a certain type of reducibility property. This property allows us to ``lift'' $S$ to a three-term progression free subset of $\mathbb{Z}_m^n$ for large $n$ (even though the original set $S\subset \mathbb{Z}_m^2$ does contain three-term arithmetic progressions).

Figures (1)

• Figure 3.1: The set $T$ defined in Definition \ref{['defi-T']}

Theorems & Definitions (24)

• Theorem 1.1
• Theorem 1.2
• Definition 2.1
• Definition 2.2
• Definition 2.3
• Proposition 2.4
• Proposition 2.5
• proof : Proof of Theorem \ref{['thm-main-2']}, assuming Propositions \ref{['proposition-1']} and \ref{['proposition-2']}
• proof : Proof of Proposition \ref{['proposition-1']} for odd $m$
• proof : Proof of Proposition \ref{['proposition-1']} for even $m$