On gaps between sums of four fourth powers

TL;DR

This paper proves that for almost all large numbers, there exists a sum of four fourth powers within a short interval just below the number, establishing a new bound on the distribution of such sums.

Contribution

It introduces a new bound on the interval length where sums of four fourth powers can be found for almost all large integers.

Findings

Almost all large integers have a sum of four fourth powers within a short interval below them.

Established the bound gd=0.24774.. for the interval length.

The result advances understanding of the distribution of sums of four fourth powers.

Abstract

We prove that for almost all $N$ there is a sum of four fourth powers in the interval $(N-N^\gamma,N]$, for all $\gamma>4059/16384=0.24774..$.