Intermediate interaction-order scale

Characterize the asymptotic behavior of the interaction constants K_{m,M} in the intermediate regime sqrt(m/log m) << M << m, between the presently known contractive upper-bound range and the linear-density noncontractive lower-bound range.

Background

For polynomials whose monomials involve at most M variables, the paper proves an upper estimate implying K_{m,M}→1 when M=o(sqrt(m/log m)), with uniform boundedness at the borderline scale M asymp sqrt(m/log m).

The entropy construction yields noncontractivity for certain interaction densities M proportional to m. The authors explicitly state that the behavior between these regimes is unresolved, identifying a gap between support-recovery methods on the upper side and entropy-producing constructions on the lower side.