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Aug 31, 2020 at 0:27 comment added Will Sawin @GeoffreyIrving Every connected set with more than two points has Hausdorff dimension at least $1$, since it projects to an interval with dimension 1. This suggests that every similar idea with a connected set will be an overkill.
Aug 30, 2020 at 21:29 comment added Geoffrey Irving Nice! Though intuitively, it feels like this doesn’t quite correspond to the notion of a “thickening”, since it maps each point of a disconnected set. Not sure how to formalize that, however.
Aug 30, 2020 at 20:02 history answered Yuval Peres CC BY-SA 4.0