Davenport's method and slim exceptional sets: the asymptotic formulae in Waring's problem

TL;DR

This paper improves estimates for exceptional sets in Waring's problem using Davenport's method, demonstrating that the asymptotic formula for seven cubes holds for almost all natural numbers up to N.

Contribution

It applies Davenport's method to refine bounds on slim exceptional sets in Waring's problem, establishing the formula's validity for nearly all relevant numbers.

Findings

Asymptotic formula holds for all but O(N^{1/3+ε}) numbers

Improved estimates for exceptional sets in Waring's problem

Validation of the formula for sums of seven cubes

Abstract

We apply a method of Davenport to improve several estimates for slim exceptional sets associated with the asymptotic formula in Waring's problem. In particular, we show that the anticipated asymptotic formula in Waring's problem for sums of seven cubes holds for all but $O(N^{1/3+\epsilon})$ of the natural numbers not exceeding $N$.