$L^p$ Asymptotics of the M\"obius Energy Density of Helix Curves

Motivated by the recent work of Lipton on the Möbius energy of helix curves, we extend the study to the $L^p$ asymptotics of the meromorphic family \[ M_\rho(t) = \frac{\rho^2+1}{\rho^2 t^2 + 4 \sin^2(t/2)} - \frac{1}{t^2}. \] The helix has infinite Möbius energy, but the arclength-rescaled energy density is finite.

As $\rho \to 0$ the helix coils infinitely tight.

Using contour integration and a careful Laurent expansion near the poles, we establish $I_p(\rho) := \left(\int_{-\infty}^\infty M_\rho(t)^p \, dt\right)^{1/p} \sim C_p \, \rho^{-(2-1/p)} $ for integer $p > 1$, extended to real $p > 1$, where $C_p$ is an explicit constant involving $\zeta(2p-1)$.

The result gives the precise $L^p$ blowup rate of the Möbius energy density as the pitch $\rho \to 0$.

The borderline case $p=1$ yields a logarithmic correction $I_1(\rho) \sim \log(1/\rho)/\rho$, recovering Lipton's main theorem.

We derive a quantitative coiling barrier and establish bilipschitz regularity for non-coiling helices.

Numerical verification confirms the scaling exponent to high precision.