Sums of cubes with shifts

Let μ₁,…,μ_s be real numbers, with μ₁ irrational. We investigate sums of shifted cubes F(x₁,…,x_s) = (x₁−μ₁)³ + ⋯ + (x_s−μ_s)³. We show that if η is real, τ>0 is sufficiently large, and s⩾9, then there exist integers x₁>μ₁,…,x_s>μ_s such that |F(x)−τ|<η. 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.