Sums of cubes with shifts

Abstract

Let ${\mu }_1,\dots ,{\mu }_s$ be real numbers, with ${\mu }_1$ irrational. We investigate sums of shifted cubes $F(x_1,\dots ,x_s)=(x_1-{\mu }_1{)}^3+\dots +(x_s-{\mu }_s{)}^3$. We show that if $\eta$ is real, $\tau >0$ is sufficiently large, and $s\ge 9$, then there exist integers $x_1>{\mu }_1,\dots ,x_s>{\mu }_s$ such that $|F(\mathbf{x})-\tau |<\eta$. This is a real analogue to Waring's problem. We then prove a full density result of the same flavour for $s\ge 5$. For $s\ge 11$, we provide an asymptotic formula. If $s\ge 6$ then $F({\mathbf{Z}}^s)$ is dense on the reals. Given nine variables, we can generalise this to sums of univariate cubic polynomials.