Let N be the set of all nonnegative integers. For any positive
integer k and any subset A of nonnegative integers, let r1,k(A,n) be
the number of solutions (a1,a2) to the equation n=a1+ka2. In 2016, Qu
proved that n→∞liminfr1,k(A,n)=∞ providing that
r1,k(A,n)=r1,k(N∖A,n) for all sufficiently large
integers, which answered affirmatively a 2012 problem of Yang and Chen. In a
very recent article, another Chen (the first named author) slightly improved
Qu's result and obtained that
n→∞liminflognr1,k(A,n)>0. In this note, we
further improve the lower bound on r1,k(A,n) by showing that
n→∞liminfnr1,k(A,n)>0. Our bound reflects
the correct order of magnitude of the representation function r1,k(A,n)
under the above restrictions due to the trivial fact that $r_{1,k}(A,n)\le
n/k.