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

Abstract

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 $k\ge 4$.

18pp

References in corpus (1)

• Multigrade efficient congruencing and Vinogradov's mean value theorem

Cited by in corpus (2)

• Rational lines on cubic hypersurfaces
• Waring-Goldbach problem in short intervals