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ADDITIVE AND SUBTRACTIVE BASES OF
$\mathbb {Z}_m$ IN AVERAGE
- Part of
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- Journal:
- Bulletin of the Australian Mathematical Society , First View
- Published online by Cambridge University Press:
- 25 November 2024, pp. 1-10
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- Article
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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$.
