Subsets of free groups with distinct differences

TL;DR

This paper investigates the maximum size of subsets within free groups that have pairwise distinct differences, establishing asymptotic bounds based on the group's rank and diameter constraints.

Contribution

It introduces the concept of distinct difference configurations in free groups and determines their maximal size asymptotically for fixed rank and large diameter.

Findings

Maximum size of such subsets is approximately (2n-1)^{d/3} for fixed n and large d.

Provides bounds on the size of subsets with distinct differences in free groups.

Analyzes the structure of free groups to derive these asymptotic results.

Abstract

Let $F_n$ be a free group of rank $n$, with free generating set $X$. A subset $D$ of $F_n$ is a \emph{Distinct Difference Configuration} if the differences $g^{-1}h$ are distinct, where $g$ and $h$ range over all (ordered) pairs of distinct elements of $D$. The subset $D$ has diameter at most $d$ if these differences all have length at most $d$. When $n$ is fixed and $d$ is large, the paper shows that the largest distinct difference configuration in $F_n$ of diameter at most $d$ has size approximately $(2n-1)^{d/3}$.