A remark on densities of hyperbolic dimensions for conformal iterated function systems with applications to conformal dynamics and fractal number theory

In this note we investigate radial limit sets of arbitrary regular conformal iterated function systems. We show that for each of these systems there exists a variety of finite hyperbolic subsystems such that the spectrum made of the Hausdorff dimensions of the limit sets of these subsystems is dense in the interval between 0 and the Hausdorff dimension of the given conformal iterated function system. This result has interesting applications in conformal dynamics and elementary fractal number theory.