# Difference bases in cyclic groups

## TL;DR

This paper investigates the minimal size of difference bases in cyclic groups, establishing bounds and exact values for groups of small order, advancing understanding of their combinatorial structure.

## Contribution

It provides explicit bounds for the difference size of cyclic groups and computes exact values for groups up to order 100, improving previous knowledge.

## Key findings

- Bounds for difference size: (1+√(4|n|-3))/2 ≤ Δ[C_n] ≤ 1.5√n
- Improved bounds for large n: Δ[C_n] ≤ (12/√73)√n for n ≥ 9 and n ≥ 2×10^15
- Exact difference sizes calculated for all cyclic groups of order ≤ 100

## Abstract

A subset $B$ of an Abelian group $G$ is called a difference basis of $G$ if each element $g\in G$ can be written as the difference $g=a-b$ 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 $\Delta[G]$. We prove that for every $n\in\mathbb N$ the cyclic group $C_n$ of order $n$ has difference size $\frac{1+\sqrt{4|n|-3}}2\le \Delta[C_n]\le\frac32\sqrt{n}$. If $n\ge 9$ (and $n\ge 2\cdot 10^{15}$), then $\Delta[C_n]\le\frac{12}{\sqrt{73}}\sqrt{n}$ (and $\Delta[C_n]<\frac2{\sqrt{3}}\sqrt{n}$). Also we calculate the difference sizes of all cyclic groups of cardinality $\le 100$.