Waring’s problem with shifts

Sam Chow

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

6 Citations (Scopus)
217 Downloads (Pure)


Let µ1, . . . , µs be real numbers, with µ1 irrational. We investigate sums of shifted kth powers F(x1, . . . , xs) = (x1−µ1)k+. . .+(xs−µs) k. For k > 4, we bound the number of variables needed to ensure that if η is real and τ > 0 is sufficiently large then there exist integers x1 > µ1, . . . , xs > µs such that |F(x) − τ | < η. This is a real analogue to Waring’s problem. When s > 2k 2 − 2k + 3, we provide an asymptotic formula. We prove similar results for sums of general univariate degree k polynomials.
Original languageEnglish
Pages (from-to)13-46
Number of pages33
Issue number1
Early online date6 Mar 2015
Publication statusPublished - Jan 2016


  • Diophantine inequalities
  • forms in many variables
  • inhomogeneous polynomials


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