# Improved Lower Bound for Difference Bases

## TL;DR

This paper improves the lower bound on the size of difference bases in integers by applying Fourier analysis, showing previous bounds were not optimal and advancing understanding in additive number theory.

## Contribution

It introduces Fourier-analytic methods to establish a sharper lower bound for the minimal size of difference bases, surpassing previous bounds.

## Key findings

- New lower bound on difference bases exceeds previous 1.5602...
- Fourier analysis proves previous bounds are not sharp
- Advances theoretical understanding of difference bases

## Abstract

A difference basis with respect to $n$ is a subset $A \subseteq \mathbb{Z}$ such that $A - A \supseteq \{1, \ldots, n\}$. R\'{e}dei and R\'{e}nyi showed that the minimum size of a difference basis with respect to $n$ is $(c+o(1))\sqrt{n}$ for some positive constant $c$. The best previously known lower bound on $c$ is $c \geqslant 1.5602\ldots$, which was obtained by Leech using a version of an earlier argument due to R\'{e}dei and R\'{e}nyi. In this note we use Fourier-analytic tools to show that the Leech--R\'{e}dei--R\'{e}nyi lower bound is not sharp.