Waring’s problem with shifts

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.