The large $k$-term progression-free sets in ${\mathbb{Z}}_q^n$

Abstract

Let $k$ and $n$ be fixed positive integers. For each prime power $q⩾k⩾3$, we show that any subset $A\subseteq {\mathbb{Z}}_q^n$ free of $k$-term arithmetic progressions has size $|A|⩽c_k(q{)}^n$ with a constant $c_k(q)$ that can be expressed explicitly in terms of $k$ and $q$. As a consequence, we can take $c_k(q)=0.8415q$ for sufficiently large $q$ and arbitrarily fixed $k\ge 3$.

References in corpus (1)

• Progression-free sets in Z_4^n are exponentially small