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

TL;DR

This paper investigates the minimal average size of additive and subtractive bases in residue class groups, improving the upper bound on the limit superior of this average from 192 to 144.

Contribution

It establishes a tighter upper bound of 144 for the limit superior of the average size of additive bases in residue groups, advancing previous results.

Findings

Upper bound of 144 for the limit superior of average base size

Improved upon previous bound of 192 by Ding and Zhao

Parallel results obtained for subtractive bases

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 $\sigma_A(n)\geq1$ for all $n\in \mathbb{Z}_m$. Let $$ \ell_m=\min\limits_{A\in \mathcal{H}_m}\left\lbrace m^{-1}\sum_{n\in \mathbb{Z}_m}\sigma_A(n)\right\rbrace. $$ Following a prior result of Ding and Zhao on Ruzsa's number, we know that $$ \limsup_{m\rightarrow\infty}\ell_m\le 192. $$ Ding and Zhao then asked possible improvements on this value. In this paper, we prove $$ \limsup\limits_{m\rightarrow\infty}\ell_m\leq 144. $$ Moreover, parallel results on subtractive bases of $ \mathbb{Z}_m$ were also investigated here.