Badly approximable points are those for which Dirichlet’s approximation theorem cannot be improved by more than a constant. While the Lebesgue measure of badly approximable points is zero, the points are still plentiful since their Hausdorff dimension is full. Moreover, the badly approximable points should be very uniformly spread out, and in particular it is expected that Hausdorff dimension of badly approximable points on fractal sets, which are not designed to specifically avoid them, should be equal to the Hausdorff dimension of the fractal. This phenomenon has been verified for many classical fractals, such as self-similar and self-conformal sets, as well as for some self-affine carpets. I will discuss a joint work with Jonathan Fraser and Henna Koivusalo, where we provide the first examples of attractors of non-linear, non-conformal iterated function systems, where this phenomenon is present. The class of attractors we consider can be thought of as non-linear analogues of self-affine Baranski carpets, in particular, we do not impose a dominating direction for the non-conformal mappings in the iterated function system.
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