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## Main Question or Discussion Point

My question, because I keep seeing this on the internet, is that if S is a subset of R and Hausdorff dimension greater than 0, it is uncountable... is this true.

It seems not to be. If one were to modify the Cantor third set and remove some length of 1/n from the middle of the sets at each iteration, one would achieve a set with Hausdorff dimension: 2 = n^d => ln2/ln n =d, and as 1/n -> 1 , n -> infinity, and ln2/ln n -> 0. Yet the set is still uncountable.

Perhaps I have missed something, or perhaps most of the time the relationship holds. Does anyone know either way? or if there are only special cases where this happens, what they are?

Thanks,

jon

It seems not to be. If one were to modify the Cantor third set and remove some length of 1/n from the middle of the sets at each iteration, one would achieve a set with Hausdorff dimension: 2 = n^d => ln2/ln n =d, and as 1/n -> 1 , n -> infinity, and ln2/ln n -> 0. Yet the set is still uncountable.

Perhaps I have missed something, or perhaps most of the time the relationship holds. Does anyone know either way? or if there are only special cases where this happens, what they are?

Thanks,

jon