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WARING’S PROBLEM WITH SHIFTS

  • Sam Chow (a1)

Abstract

Let ${\it\mu}_{1},\ldots ,{\it\mu}_{s}$ be real numbers, with ${\it\mu}_{1}$ irrational. We investigate sums of shifted $k$ th powers $\mathfrak{F}(x_{1},\ldots ,x_{s})=(x_{1}-{\it\mu}_{1})^{k}+\cdots +(x_{s}-{\it\mu}_{s})^{k}$ . For $k\geqslant 4$ , we bound the number of variables needed to ensure that if ${\it\eta}$ is real and ${\it\tau}>0$ is sufficiently large then there exist integers $x_{1}>{\it\mu}_{1},\ldots ,x_{s}>{\it\mu}_{s}$ such that $|\mathfrak{F}(\mathbf{x})-{\it\tau}|<{\it\eta}$ . This is a real analogue to Waring’s problem. When $s\geqslant 2k^{2}-2k+3$ , we provide an asymptotic formula. We prove similar results for sums of general univariate degree- $k$ polynomials.

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