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

TL;DR

This paper proves that for almost all large numbers, there exists a sum of two positive integral cubes within a specific short interval, and extends similar results to higher powers, improving understanding of additive representations.

Contribution

It establishes new bounds for the existence of sums of two like powers in short intervals, refining previous results for cubes and higher powers.

Findings

Almost all large numbers have a sum of two positive cubes in a short interval.

Results extend to sums of two positive integral k-th powers for k ≥ 4.

Improves bounds between trivial and known constraints for additive representations.

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 $k\ge 4$.