Polynomial control of the NUQFT geometry parameter for NUCT nodes

Determine a closed-form or polynomial-in-N upper bound for the geometry parameter kappa associated with the non-uniform quantum Fourier transform when its nodes are the angular values t_k=(2pi)^{-1}\arccos(-1+2k/N) of the non-uniform Chebyshev transform.

Background

The prior NUQFT analysis depends on the geometry parameter $\kappa$, which measures the proximity of scaled node offsets to the boundaries $\pm1$ of their nearest-grid cells. For the NUCT nodes, controlling $\kappa$ is equivalent to obtaining a lower bound on $\delta_N=1-\max_k|y_k|$, where $y_k$ is the node offset.

The paper derives only an exponentially small lower bound on $\delta_N$, leading to a potentially exponential complexity bound for the earlier NUQFT method. It notes that a polynomial bound on $\kappa$ would require controlling the distance of certain quantities involving $\arccos(-1+2k/N)$ from half-integers uniformly in $k$ and $N$, a problem for which the authors report no satisfactory argument. The conditioning-free NUQFT developed in the paper avoids requiring this bound, but does not resolve the underlying number-theoretic question.