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 $\sigma _A(n)\geq 1$ for all $n\in \mathbb {Z}_m$ . Let $$ \begin{align*} \ell_m=\min\limits_{A\in \mathcal{H}_m}\bigg\lbrace m^{-1}\sum_{n\in \mathbb{Z}_m}\sigma_A(n)\bigg\rbrace. \end{align*} $$ Ding and Zhao [‘A new upper bound on Ruzsa’s numbers on the Erdős–Turán conjecture’, Int. J. Number Theory 20 (2024), 1515–1523] showed that $\limsup _{m\rightarrow \infty }\ell _m\le 192$ . We prove $$ \begin{align*} \limsup\limits_{m\rightarrow\infty}\ell_m\leq 144 \end{align*} $$ and investigate parallel results on subtractive bases of $ \mathbb {Z}_m$ .