Improvement of cumulant bounds for admissible unbounded symbols

Determine whether the super-exponential cumulant bound for admissible symbols with potentially unbounded positive and negative Fourier parts can be substantially improved in full generality, or whether such an improvement requires additional assumptions.

Background

For admissible symbols satisfying the Fourier-coefficient decay condition but without boundedness of the positive and negative Fourier parts, the paper proves bounds of the form $\lvert C_m^{(n)}(f)\rvert\leq a( cm)^m$ for cumulants of order $m\geq 3$. These estimates are uniform in $n$, but grow super-exponentially in the cumulant order.

The authors note that the Fisher–Hartwig setting suggests that this growth is far from optimal. A sharper bound analogous to the exponential-in-order estimate established for symbols with bounded decompositions would yield stronger control of the determinant expansion, but the paper does not determine whether such an improvement is valid for all admissible symbols or only for narrower subclasses.