On the asymptotic formula in Waring's problem with shifts

TL;DR

This paper establishes an asymptotic formula for the count of solutions to a shifted Waring's problem for integers with large parameters, using advanced analytic number theory techniques.

Contribution

It introduces a new estimate on Hardy--Littlewood minor arcs and applies Freeman’s Davenport--Heilbronn method to improve existing results for Waring's problem with shifts.

Findings

Asymptotic formula proven for s ≥ k^2 + (3k-1)/4

New estimate on Hardy--Littlewood minor arcs developed

Extension of results to irrational shifts in Waring's problem

Abstract

We show that for integers $k\geq 4$ and $s\geq k^2+(3k-1)/4$, we have an asymptotic formula for the number of solutions, in positive integers $x_i$, to the inequality $\left|(x_1-\theta_1)^k+\dotsc+(x_s-\theta_s)^k-\tau\right|<\eta$, where $\theta_i\in(0,1)$ with $\theta_1$ irrational, $\eta\in(0,1]$, and $\tau>0$ is sufficiently large. We use Freeman's variant of the Davenport--Heilbronn method, along with a new estimate on the Hardy--Littlewood minor arcs, to obtain this improvement on the original result of Chow.