The lower bound for difference bases

Abstract

A set $A\\subseteq~\\mathbb{Z}$ is called a difference basis with respect to $n$ if $A-A\\supseteq~\\{1,~2,~\\ldots,~n\\}$. The minimum size of a difference basis, while it is a natural combinatorial number theory question in its own right, also hasapplications to graceful labelings of graphs, to symmetric intersecting families ofsets, and to signal processing. It is closely and deeply related to the theory of cryptography and coding theory. Applying the method of Fourier analysis, we discuss the properties of some more advanced Fourier coefficients of parameters, further improve the lower bound of the difference basis, therefore improve the related results of Rédei and Rényi (1949), Leech (1956), and Bernshteyn and Tait (2019).