Crystal realization of arithmetic obstructions to dispersion

Determine whether the step-function Bernoulli-convolution construction can be perturbed or smoothed to produce a genuinely long-range crystal without dispersion while retaining the associated arithmetic obstruction.

Background

The authors consider a discontinuous step-function phase whose occupation measure is a Bernoulli convolution. For reciprocal parameters associated with Pisot numbers, the Fourier transform of that measure does not tend to zero, indicating an arithmetic obstruction to dispersion. Because the discontinuous phase does not define a crystal in the class studied, the unresolved issue is whether smoothing or perturbing it can preserve the obstruction while yielding a valid long-range crystal.

References

This phase does not itself define a crystal of the class considered here, but it raises a natural question: can one perturb or smooth this construction to obtain a genuinely long-range crystal without dispersion while retaining the arithmetic obstruction?

— Arbitrarily Fast Quantum Dispersion in Long-Range Crystals  (2608.27326 - Leclerc et al., 27 Aug 2026) in Section 3, subsection “Prospects for the theory,” part (a) “Long-range crystals without dispersion”