McMullen's game for equicontinuously-twisted badly approximable points in continued fractions and beta expansions
David Lambert, David Simmons, Jiajie Zheng
TL;DR
The paper investigates twisted badly approximable sets arising from equicontinuous twists in continued fraction and beta-expansion dynamics. By developing inductive McMullen-style strategies using a combinatorial framework of n-th order vertices and level-types, it proves that the twisted set is β-absolute winning for all β in (0,1/3) in both the beta-transformation (System I) and Gauss map (System II). This yields full Hausdorff dimension and α-winning consequences, significantly extending prior results for constant twists. The approach offers a robust, structure-based method for establishing strong fractal and game-theoretic properties of recurrence/shrinking-target sets under equicontinuous perturbations.
Abstract
In a beta-transformation (for integer beta) or a Gauss map system, given a sequence of functions fn from [0,1] to itself, consider the collection of points in [0,1] whose nth iteration under the map is distanced away from its value under fn. It is well known that for constant sequences fn, such collections are always winning in McMullen's game and in particular they have Hausdorff dimension 1. We extend the results to all equicontinuous sequences of functions fn.