Full inverse-sine rigidity conjecture

Determine whether the full inverse-sine rigidity principle proposed by Graham and O'Bryant holds beyond the subcritical regime in which every representative satisfies 3a<Q, including whether the corresponding inverse-sine row system must contain a deficient row without the cardinality-dependent condition Q>(7/4)^{|A|}.

Background

The paper studies finite row systems formed from distinct units modulo a denominator Q, with interaction weights given by the inverse-sine kernel K_Q(d)=sin(π/Q)/sin(πd/Q). In the application to Beatty partitions, Fourier cancellation forces every row in a minimum-gcd layer to have mass at least one. The paper proves a subcritical result: when all representatives satisfy 3a<Q and their total is at most Q, at least one row has mass below one.

The authors identify this theorem as only a subcritical version of a broader rigidity principle associated with Graham and O'Bryant. Their result removes a cardinality-dependent hypothesis within the subcritical regime, but explicitly does not claim the full inverse-sine conjecture. Thus the unrestricted validity of the broader principle remains unresolved in the scope of the paper.