McMullen's game for equicontinuously-twisted badly approximable points in continued fractions and beta expansions

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.