@Claus - with the Buffon noodle, we take the original problem referring to a unit-length, straight needle, and we generalize in two directions: 1) to arbitrary length, and 2) to a curved needle ("noodle"). Neither of these is formally generalizing to higher dimension or higher cardinality - we just allow the needle to be longer (so, a finer notion than cardinality is being generalized here) and, if you choose to look at it this way, we give it "one more dimension" of wiggle room. I was just wondering what your point of view on this example is - does it fit your intended criteria? :)

Do you count Buffon's noodle as an example of this? en.wikipedia.org/wiki/Buffon%27s_noodle

just a little note: $BA^{-1}$ is not necessarily symmetric - was that the intention? If not, maybe you can replace it by something like $A^{-1/2}BA^{-1/2}$ (of course this still works fine but you need to define the log without appealing to the diagonalization)

you kind of have to right? otherwise the original question is false too: e.g. if all the points are on a line

@SalvoTringali yes, I'm aware of this - though the proofs I've heard about mentioned cyclotomic polynomials. Though I guess for the case of a prime difference the cyclotomic polynomials are related to the Chebyshev polynomials? Anyway. The dream was to somehow deduce the cases of Dirichlet we need to disprove the existence of a polynomial by first assuming its (false) existence :) but I haven't been able to get any more interesting cases ;/