On the representation of an exponential type sequence

Let $p,q＞1$ be integers with $(p,q)=1$. In 1959, Birch proved that every sufficiently large integer $n$ can be represented as a sum of distinct integers of the form $p^\alpha q^\beta$. In this paper, we shall prove that for any real number $\varepsilon＞0$, there is a positive real number $c=c(p,q,\varepsilon)$ such that every sufficiently large integer $n$ can be represented as a sum of distinct integers of the form $p^\alpha q^\beta$, all of which are greater than $\displaystyle\frac{cn}{(\log n)^{1+\varepsilon}}$.