Note on a theorem of Birch–Erdős and m-ary partitions

Let $p,q>1$ be two relatively prime integers and $N$ the set of nonnegative integers. Let $f_{p,q}\left(n\right)$ be the number of different expressions of n written as a sum of distinct terms taken from $\{p^{\alpha }{q}^{\beta }:\alpha ,\beta \in N\}$. Erdős conjectured and then Birch proved that $f_{p,q}\left(n\right)\ge 1$ provided that n is sufficiently large. In this note, for all sufficiently large number n we prove$f_{p,q}\left(n\right)=2^{\frac{{(\log n)}^2}{2\log p\log q}(1+O(\log \log n/\log n\left)\right)}.$ We also show that ${\lim }_{n\to \infty }{f}_{2,q}(n+1)/f_{2,q}\left(n\right)=1$. Additionally, we will point out the relations between $f_{2,q}\left(n\right)$ and m-ary partitions.