Optimal error rates in the entropy lower bounds

Determine the optimal rates, as functions of the high-entropy parameter M and the prime-field entropy deficit K, of the error terms in the torsion-free and prime-cyclic high-entropy entropy bounds.

Background

The main theorem supplies explicit error terms that vanish as the larger input entropy M tends to infinity and, in prime cyclic groups, as the deficit K between the uniform entropy and M also tends to infinity. The proof yields terms of order 1/M and, in the prime-field case, additional terms of order approximately (log K)/K.

The authors explicitly leave unresolved whether these rates are optimal. Establishing optimal rates would sharpen the quantitative form of the asymptotic half-bit theorem.