Exact transition behavior after the first analytic branch departure

Establish an exact transition equation at fixed contraction parameter M for the intermediate dimension curve of the delayed-contraction Cantor set E_{\beta,M}, characterize the optimal digit-frequency profile after departure from the continued analytic profile, and determine the differentiability of the intermediate dimension curve at the transition.

Background

The paper studies the intermediate dimension curve d_\theta of the delayed-contraction Cantor set E_{\beta,M}. It derives a finite pressure characterization and proves a sharp asymptotic separation between the first departure from a continued analytic profile and the collision of that profile’s frequency intervals, particularly for large M.

The authors do not resolve the exact fixed-M transition mechanism. The unresolved issues concern the precise transition equation, the optimizer once the continued four-frequency profile ceases to be optimal, and the regularity of d_\theta at the transition point.