The dimension spectrum of the infinitely generated Apollonian gasket

Abstract 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.