Waring's problem with shifts

Abstract

Let ${\mu }_1,\dots ,{\mu }_s$ be real numbers, with ${\mu }_1$ irrational. We investigate sums of shifted $k$th powers $\mathfrak{F}(x_1,\dots ,x_s)=(x_1-{\mu }_1{)}^k+\dots +(x_s-{\mu }_s{)}^k$. 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 $x_1>{\mu }_1,\dots ,x_s>{\mu }_s$ such that $|\mathfrak{F}(\mathbf{x})-\tau |<\eta$. 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.

References in corpus (1)

• Sums of cubes with shifts

Cited by in corpus (2)

• Arithmetic Ramsey theory over the primes
• Almost equal summands in Waring's problem with shifts