Spectral refinement for sparse sequences

Develop a decomposition for sparse polynomial averages—such as along primes or suitable Hardy sequences—analogous to the spectral refinement of the Bergelson–Host–Kra decomposition, with the structured nilsequence part having spectrum contained in the spectrum of the underlying transformation.

Background

Moreira and Richter proved a spectral refinement of the Bergelson–Host–Kra decomposition for consecutive iterates.

Extending such a refinement to sparse sequences would significantly advance structural understanding of sparse multiple averages.