Sums of cubes with shifts

Let mu(1), ... , mu(s) be real numbers, with mu(1) irrational. We investigate sums of shifted cubes F(x(1), ... , x(s)) = (x(1) - mu(1))(3) + center dot center dot center dot + (x(s) - mu(s))(3). We show that if eta is real, tau > 0 is sufficiently large, and s >= 9, then there exist integers x(1) > mu(1), ... , x(s) > mu(s) such that vertical bar F(x) -tau vertical bar < eta. This is a real analogue to Waring's problem. We then prove a full density result of the same flavour for s >= 5. For s >= 11, we provide an asymptotic formula. If s >= 6, then F(Z(s)) is dense on the reals. Given nine variables, we can generalize this to sums of univariate cubic polynomials.