Let S={12,22,32,...} be the set of squares and
W={wnβ}n=1βββN be an additive
complement of S so that S+Wβ{nβN:nβ₯N0β} for some N0β. Let
RS,Wβ(n)=#{(s,w):n=s+w,sβS,wβW}.
In 2017, Chen-Fang \cite{C-F} studied the lower bound of
βn=1NβRS,Wβ(n). In this note, we improve
Cheng-Fang's result and get that
n=1βNβRS,Wβ(n)βNβ«N1/2. As an application,
we make some progress on a problem of Ben Green problem by showing that
nββlimsupβn16Ο2βn2βwnβββ₯4Οβ+80.193Ο2β.Comment: The new version significantly improves the result of the former on