# Difference bases in dihedral groups

## TL;DR

This paper investigates the minimal size of difference bases in dihedral groups, establishing bounds on their difference characteristic and computing exact values for small groups, advancing understanding of their algebraic structure.

## Contribution

It provides new bounds on the difference characteristic of dihedral groups and computes exact difference sizes for groups up to order 80.

## Key findings

- For all n, the difference characteristic of D_{2n} is between √2 and approximately 1.983.
- For large n (≥ 2×10^{15}), the difference characteristic is less than approximately 1.633.
- Exact difference sizes and characteristics are computed for all dihedral groups of order ≤ 80.

## Abstract

A subset $B$ of a group $G$ is called a difference basis of $G$ if each element $g\in G$ can be written as the difference $g=ab^{-1}$ of some elements $a,b\in B$. The smallest cardinality $|B|$ of a difference basis $B\subset G$ is called the difference size of $G$ and is denoted by $\Delta[G]$. The fraction $\eth[G]:=\Delta[G]/{\sqrt{|G|}}$ is called the difference characteristic of $G$. We prove that for every $n\in\mathbb N$ the dihedral group $D_{2n}$ of order $2n$ has the difference characteristic $\sqrt{2}\le\eth[D_{2n}]\leq\frac{48}{\sqrt{586}}\approx1.983$. Moreover, if $n\ge 2\cdot 10^{15}$, then $\eth[D_{2n}]<\frac{4}{\sqrt{6}}\approx1.633$. Also we calculate the difference sizes and characteristics of all dihedral groups of cardinality $\le80$.