Improving dependence on frequency bounds and approximation error

Improve the dependence on the frequency cap and approximation parameter in the paper’s algorithms for estimating F_k with k in (0,1) and related Bernstein-type statistics, potentially by handling singularities more effectively or using targeted or adaptive quadrature methods.

Background

The algorithms developed in the paper have polynomial dependence on the frequency bound and the inverse approximation parameter. The authors identify this dependence as a principal efficiency limitation and suggest that more refined treatment of the singularities or more specialized numerical integration could improve the bounds.

The problem is explicitly unresolved because the paper asks whether these dependencies can be reduced for both F_k and the other statistics studied.