  The heavy jet mass event shape has large perturbative logarithms near the
leading order kinematic threshold at $\rho = \frac{1}{3}$. Catani and Webber
named these logarithms Sudakov shoulders and resummed them at
double-logarithmic level. A resummation to next-to-leading logarithmic level
was achieved recently. Here, we extend the resummation using an effective field
theory framework to next-to-next-to-leading logarithmic order and show how to
combine it with the resummation of dijet logarithms. We also solve the open
problem of an unphysical singularity in the resummed momentum space
distribution, in a way similar to how it is resolved in the Drell-Yan $q_T$
spectrum: through a careful analysis of the kinematics and scale-setting in
position space. The heavy jet mass Sudakov shoulder is the first observable
that does not involve transverse momentum for which position space resummation
is critical. These advances may lead to a more precise extraction of the strong
coupling constant from $e^+ e^-$ data.
