Additive representation in short intervals, II: sums of two like powers

We establish that, for almost all natural numbers $N$, there is a sum of two positive integral cubes lying in the interval $[N-N^{7\mathrm{/}18+ϵ},N]$. Here, the exponent $7\mathrm{/}18$ lies half way between the trivial exponent $4\mathrm{/}9$ stemming from the greedy algorithm, and the exponent $1\mathrm{/}3$ 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 $k\ge 4$.