Amended Loneliness Spectrum Conjecture

Determine whether for every integer n ≥ 1 and every n-set V of positive integers, either κ(V) equals s/(sn + k) for some s,k ∈ N (almost tight instances), or κ(V) ≥ 1/n (uniform lower bound away from almost tight values).

Background

Kravitz originally conjectured a discrete spectrum for possible loneliness gaps, which was later disproved for some n. Fan and Sun proposed an amended conjecture that broadens the family of ‘almost tight’ values while maintaining a uniform gap for non-almost-tight sets.

Recent work has studied spectra relative to subtori and identified rigid arithmetic structures, offering partial insights for specific n and intervals.