[Cross-posted from Math.SE because I got no responses there.]

Given a polynomial in $e^{\mathrm{i}k t}$ of the form $$ p(t) = \sum_{-n\leq k\leq n} c_k e^{\mathrm{i}k t} $$ with $\bar c_{-k} = c_k$, is there a good way of characterising how close its roots can be in terms of its coefficients?

Clearly this depends on coefficients, for example $1-\epsilon-\cos x$ has roots $\sqrt{\epsilon}$ apart, but for general coefficients, is there any good result? I am secretly hoping for minimum distance of something like $A n^{-1}$ (which is the case for $\sin n x$) for non-almost-degenerate coefficients.

If you know a result for any kind of orthogonal polynomials (like Chebyshev polynomials), that would also be very helpful. Also, I know there are some results for asymptotic distributions of eigenvalues of Toeplitz matrices, but I am interested in a result for a fixed polynomial $p$ and its coefficients, and not just $n\to\infty$.