Waring's problem with shifts

Abstract

Let $μ_1, \ldots, μ_s$ be real numbers, with $μ_1$ irrational. We investigate sums of shifted $k$th powers $\mathfrak{F}(x_1, \ldots, x_s) = (x_1 - μ_1)^k + \ldots + (x_s - μ_s)^k$. For $k \ge 4$, we bound the number of variables needed to ensure that if $η$ is real and $τ> 0$ is sufficiently large then there exist integers $x_1 > μ_1, \ldots, x_s > μ_s$ such that $|\mathfrak{F}(\mathbf{x}) - τ| < η$. 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.