  For fixed $m$ and $R\subseteq \{0,1,\ldots,m-1\}$, take $A$ to be the set of
positive integers congruent modulo $m$ to one of the elements of $R$, and let
$p_A(n)$ be the number of ways to write $n$ as a sum of elements of $A$.
Nathanson proved that $\log p_A(n) \leq (1+o(1)) \pi \sqrt{2n|R|/3m}$ using a
variant of a remarkably simple method devised by Erd\H{o}s in order to bound
the partition function. In this short note we describe a simpler and shorter
proof of Nathanson's bound.
