Conditioning of random cyclic-shift Krylov compressions

For the cyclic shift $C_n$ with $n \ge 3$, a starting vector $b$ uniform on
the complex unit sphere and $2 \le k < n$, the paper proves that the
compression $H_k = Q^*C_nQ$ of $C_n$ to the Krylov space
$\operatorname{span}\{b, C_nb, \dots, C_n^{k-1}b\}$ satisfies
$\mathbb{E}\,\kappa_V(H_k) \le 17n^2k$, where $\kappa_V$ is the eigenvector
condition number. By Markov's inequality, $\kappa_V(H_k) \le 1700n^2k$ with
probability at least $0.99$. In Fourier coordinates, $H_k$ is the compression
of multiplication by $z$ to polynomials of degree less than $k$, in an inner
product on the $n$th roots of unity with independent exponential weights. An
antiunitary symmetry matches left and right eigenvectors, so the norm of each
spectral projector is bounded by the sensitivity of its eigenvalue to the
weights; a bound on the total variation of rank-one root loci then bounds the
expectation. A second argument, by discriminant avoidance, gives a weaker
polynomial bound.