On Waring's Problem: Two Cubes and Two Minicubes

We establish that almost every positive integer $n$ is the sum of four cubes, two of which are at most $n^{\theta }$, as long as $\theta \ge 192\mathrm{/}869$. An asymptotic formula for the number of such representations is established when $1\mathrm{/}4<\theta <1\mathrm{/}3$.