Can you hear the shape of a Beatty sequence?

Let K(x_1,…,x_d) be a polynomial. If you are not given the real numbers \alpha_1, \alpha_2, …,\alpha_d, but are given the polynomial K and the sequence a_n=K(\floor{n\alpha_1},\floor{n\alpha_2},…,\floor{n\alpha_d}), can you deduce the values of \alpha_i? Not, it turns out, in general. But with additional irrationality hypotheses and certain polynomials, it is possible. We also consider the problem of deducing \alpha_i from the integer sequence with nested flooring (\floor{\floor{… \floor{\floor{n\alpha_1}\alpha_2}… \alpha_{d-1}}\alpha_d})_{n=1}^\infty.