Existence of deterministic Cantor sets achieving the best FUP exponent in the continuous setting

Construct a deterministic Cantor set in R for which the h-neighborhoods satisfy the fractal uncertainty principle (1.1) with the best possible exponent 1 − δ in the continuous semiclassical Fourier transform framework, or determine whether such an example exists.

Background

While various deterministic results provide small improvements over the volume bound for δ-regular sets, no deterministic examples are known in the continuous setting that achieve the conjecturally optimal exponent 1 − δ for Cantor sets.

The authors explicitly note the lack of known deterministic constructions meeting this threshold in the continuous Fourier framework, marking this as an outstanding existence problem.