On the asymptotic formula in Waring's problem with shifts

Kirsti D. Biggs

Research output: Contribution to journalArticle (Academic Journal)peer-review

1 Citation (Scopus)
221 Downloads (Pure)

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.

Original languageEnglish
Pages (from-to)353-379
Number of pages27
JournalJournal of Number Theory
Volume189
Early online date21 Feb 2018
DOIs
Publication statusPublished - 1 Aug 2018

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