Additive and subtractive bases of ${\mathbb{Z}}_m$ in average

Abstract

Given a positive integer $m$, let ${\mathbb{Z}}_m$ be the set of residue classes mod $m$. For $A\subseteq {\mathbb{Z}}_m$ and $n\in {\mathbb{Z}}_m$, let ${\sigma }_A(n)$ be the number of solutions to the equation $n=x+y$ with $x,y\in A$. Let ${\mathcal{H}}_m$ be the set of subsets $A\subseteq {\mathbb{Z}}_m$ such that ${\sigma }_A(n)\ge 1$ for all $n\in {\mathbb{Z}}_m$. Let $${\ell }_m=\underset{A\in {\mathcal{H}}_m}{\min }\left\{m^{-1}\sum _{n\in {\mathbb{Z}}_m}{\sigma }_A(n)\right\}.$$ Following a prior result of Ding and Zhao on Ruzsa's number, we know that $$\underset{m\to \infty }{\mathrm{limsup}}{\ell }_m\le 192.$$ Ding and Zhao then asked possible improvements on this value. In this paper, we prove $$\underset{m\to \infty }{\mathrm{limsup}}{\ell }_m\le 144.$$ Moreover, parallel results on subtractive bases of ${\mathbb{Z}}_m$ were also investigated here.

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