We prove that the infinitely generated Apollonian gasket has full Hausdorff dimension spectrum. Our proof, which is computer assisted, relies on an iterative technique introduced by the first three authors [Chousionis, Leykekhman, and Urbański. The dimension spectrum of conformal graph directed Markov systems. Selecta Math. (N.S.) 25 (2019), 74] and on a flexible method for rigorously estimating Hausdorff dimensions of limit sets of conformal iterated function systems, which we recently developed [Chousionis et al. Rigorous Hausdorff dimension estimates for conformal fractals. Preprint, 2024, arXiv:2408.06330]. Another key ingredient in our proof is obtaining, for the first time, reasonably sized distortion constants for the (infinite) Apollonian iterated function system.
