On Vu's restricted box estimate in Waring's problem

TL;DR

This paper improves bounds on the number of solutions to Waring's problem within a box, reducing the number of variables needed for the Hardy--Littlewood upper bound with a power-saving error.

Contribution

It demonstrates that the variable count s can be lowered to approximately k^2 - k + O(√k), improving Vu's previous bound of s ≥ O(8^k k^3).

Findings

Established a new bound s ≥ k^2 - k + O(√k) for solutions count.

Extended the applicability of Hardy--Littlewood bounds in Waring's problem.

Reduced the number of variables needed for the expected upper bound.

Abstract

In 2000, Vu proved that the number of solutions of $x_1^k + \cdots + x_s^k = N$ in an arbitrary box satisfies the expected Hardy--Littlewood upper bound with a power-saving error term, for $s \geq O(8^k k^3)$. We show that one may take $s\geq k^2 - k + O(\sqrt{k})$.