Optimal dependence on the Frostman constant

Determine the optimal dependence on the Frostman constant \(F_s\) in the intrinsic-energy estimate for angular measures \(\nu\) on the normal arc of the monomial curve \(\gamma_k(t)=(t,t^k)\), under the condition \(\nu(B(\omega,r))\le F_s\nu(\Omega_k)r^s\).

Background

The Frostman analysis yields a growth diagram for the intrinsic energy $\mathcal I_R(\nu)$, but the final uniform estimates use a factor $F_s^2$. An intermediate active-pair estimate preserves more detailed information about $F_s$, suggesting that the quadratic dependence may not be optimal. The paper explicitly leaves the problem of identifying the sharp dependence unresolved.