  We establish that, for almost all natural numbers $N$, there is a sum of two
positive integral cubes lying in the interval $[N-N^{7/18+\epsilon},N]$. Here,
the exponent $7/18$ lies half way between the trivial exponent $4/9$ stemming
from the greedy algorithm, and the exponent $1/3$ constrained by the number of
integers not exceeding $X$ that can be represented as the sum of two positive
integral cubes. We also provide analogous conclusions for sums of two positive
integral $k$-th powers when $k\ge 4$.

Number Theory