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
$$ \begin{align*} \ell_m=\min\limits_{A\in \mathcal{H}_m}\bigg\lbrace m^{-1}\sum_{n\in \mathbb{Z}_m}\sigma_A(n)\bigg\rbrace. \end{align*} $$
A result of Ding and Zhao [Reference Ding and Zhao1, Lemma 2.4] on Ruzsa’s numbers implies that
$\limsup _{m\rightarrow \infty }\ell _m\le 192$
. In [Reference Liang, Zhang and Zuo2], the authors improved this to
$\limsup _{m\rightarrow \infty }\ell _m\leq 144$
.
In the proof of [Reference Liang, Zhang and Zuo2],
$$ \begin{align} A_1=\{u+2pv:(u,v)\in B\}. \end{align} $$
As pointed by Mr. Honghu Liu, (1) implies
$|A_1|\le |B|$
rather than
$|A_1|\le 2|B|$
as stated in our article. Since
$\ell _m \le |B|^2/m$
from [Reference Liang, Zhang and Zuo2], adjusting the numbers accordingly leads to the following much stronger bound.
Theorem 1. We have
$\limsup _{m\rightarrow \infty }\ell _m\leq 36$
.
