On the asymptotic formula in Waring's problem with shifts

Kirsti D. Biggs

doi:10.1016/j.jnt.2017.12.009

On the asymptotic formula in Waring's problem with shifts

Kirsti D. Biggs

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Abstract

We show that for integers k≥4 and s≥k²+(3k−1)/4, we have an asymptotic formula for the number of solutions to the inequality |(x₁−θ₁)^k+…+(x_s−θ_s)^k−τ|<η in positive integers x_i, where θ_i∈(0,1) with θ₁ irrational, η∈(0,1], and 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.

Original language	English
Pages (from-to)	353-379
Number of pages	27
Journal	Journal of Number Theory
Volume	189
Early online date	21 Feb 2018
DOIs	https://doi.org/10.1016/j.jnt.2017.12.009
Publication status	Published - 1 Aug 2018

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title = "On the asymptotic formula in Waring's problem with shifts",

abstract = "We show that for integers k≥4 and s≥k2+(3k−1)/4, we have an asymptotic formula for the number of solutions to the inequality |(x1−θ1)k+…+(xs−θs)k−τ|<η in positive integers xi, where θi∈(0,1) with θ1 irrational, η∈(0,1], and 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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N2 - We show that for integers k≥4 and s≥k2+(3k−1)/4, we have an asymptotic formula for the number of solutions to the inequality |(x1−θ1)k+…+(xs−θs)k−τ|<η in positive integers xi, where θi∈(0,1) with θ1 irrational, η∈(0,1], and 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.

AB - We show that for integers k≥4 and s≥k2+(3k−1)/4, we have an asymptotic formula for the number of solutions to the inequality |(x1−θ1)k+…+(xs−θs)k−τ|<η in positive integers xi, where θi∈(0,1) with θ1 irrational, η∈(0,1], and 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.

UR - https://www.scopus.com/pages/publications/85041643711

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DO - 10.1016/j.jnt.2017.12.009

M3 - Article (Academic Journal)

AN - SCOPUS:85041643711

SN - 0022-314X

VL - 189

SP - 353

EP - 379

JO - Journal of Number Theory

JF - Journal of Number Theory

ER -

On the asymptotic formula in Waring's problem with shifts

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