On the asymptotic formula in Waring's problem with shifts

2016/11/29 by Biggs, Kirsti
#11D75 #11P05 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1611.09889

Abstract

We show that for integers k≥ 4 and s≥ k²+(3k-1)/4, we have an asymptotic formula for the number of solutions, in positive integers x_i, to the inequality |(x₁-θ₁)^k+\dotsc+(x_s-θ_s)^k-τ|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.

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