Sidon sets and 2-caps in $\mathbb{F}_3^n$

Abstract

For each natural number $d$, we introduce the concept of a $d$-cap in ${\mathbb{F}}_3^n$. A set of points in ${\mathbb{F}}_3^n$ is called a $d$-cap if, for each $k=1,2,\dots ,d$, no $k+2$ of the points lie on a $k$-dimensional flat. This generalizes the notion of a cap in ${\mathbb{F}}_3^n$. We prove that the $2$-caps in ${\mathbb{F}}_3^n$ are exactly the Sidon sets in ${\mathbb{F}}_3^n$ and study the problem of determining the size of the largest $2$-cap in ${\mathbb{F}}_3^n$.