Improved Lower Bound for Difference Bases

Abstract

A difference basis with respect to $n$ is a subset $A\subseteq \mathbb{Z}$ such that $A-A\supseteq \{1,\dots ,n\}$. Rédei and Rényi 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 $c⩾1.5602\dots$, which was obtained by Leech using a version of an earlier argument due to Rédei and Rényi. In this note we use Fourier-analytic tools to show that the Leech--Rédei--Rényi lower bound is not sharp.

6 pages; v2: minor changes based on referee report