Maximal line-free sets in $\mathbb{F}_p^n$

TL;DR

This paper investigates the maximum size of subsets in finite fields that contain no full line, providing new bounds and constructions especially in three dimensions, with implications for higher dimensions.

Contribution

It introduces improved bounds for the size of line-free sets in $F_p^n$, particularly in three dimensions, and extends results to higher dimensions with explicit constructions and asymptotic estimates.

Findings

Established a new lower bound for $r_p(F_p^3)$ that increases with $p$

Provided an upper bound for $r_p(F_p^3)$ and generalizations to higher dimensions

Derived asymptotic lower bounds for specific primes and dimensions

Abstract

We study subsets of $\mathbb{F}_p^n$ that do not contain progressions of length $k$. We denote by $r_k(\mathbb{F}_p^n)$ the cardinality of such subsets containing a maximal number of elements. In this paper we focus on the case $k=p$ and therefore sets containing no full line. A~trivial lower bound $r_p(\mathbb{F}_p^n)\geq(p-1)^n$ is achieved by a hypercube of side length $p-1$ and it is known that equality holds for $n\in\{1,2\}$. We will however show that $r_p(\mathbb{F}_p^3)\geq (p-1)^3+p-2\sqrt{p}$, which is the first improvement in the three dimensional case that is increasing in $p$. We will also give the upper bound $r_p(\mathbb{F}_p^{3})\leq p^3-2p^2-(\sqrt{2}-1)p+2$ as well as generalizations for higher dimensions. Finally we present some bounds for individual $p$ and $n$, in particular $r_5(\mathbb{F}_5^{3})\geq 70$ and $r_7(\mathbb{F}_7^{3})\geq 225$ which can be used to give the asymptotic lower bound $4.121^n$ for $r_5(\mathbb{F}_5^{n})$ and $6.082^n$ for $r_7(\mathbb{F}_7^{n})$.