Waring's problem with shifts

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JournalMathematika
DatePublished - Jan 2016
Issue number1
Volume62
Number of pages33
Pages (from-to)13-46
Original languageEnglish

Abstract

Let $\mu_1, \ldots, \mu_s$ be real numbers, with $\mu_1$ irrational. We investigate sums of shifted $k$th powers $\mathfrak{F}(x_1, \ldots, x_s) = (x_1 - \mu_1)^k + \ldots + (x_s - \mu_s)^k$. For $k \ge 4$, we bound the number of variables needed to ensure that if $\eta$ is real and $\tau > 0$ is sufficiently large then there exist integers $x_1 > \mu_1, \ldots, x_s > \mu_s$ such that $|\mathfrak{F}(\bx) - \tau| < \eta$. This is a real analogue to Waring's problem. When $s \ge 2k^2-2k+3$, we provide an asymptotic formula. We prove similar results for sums of general univariate degree $k$ polynomials.

    Research areas

  • diophantine inequalities, forms in many variables, inhomogeneous polynomials

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