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arXiv · 1611.09889

On the asymptotic formula in Waring's problem with shifts

Kirsti Biggs
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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.

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Kirsti Biggs. 2016-11-29. On the asymptotic formula in Waring's problem with shifts. https://arxiv.org/abs/1611.09889

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