All Cantor sets are homeomorphic, but the Hausdorff dimension of a Cantor set depends on the particular metric being considered. While there are some special situations where the Hausdorff dimension of a set can be computed easily, in general calculating Hausdorff dimension seems difficult. I was wondering, however, if the Hausdorff dimension of the set of $p$-adic integers (which forms a Cantor set) is known for each $p$, and if so what a good reference for this would be.
Hausdorff dimension of the p-adic integers
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p-adic-number-theory
fractals
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4... with respect to the $p$-adic norm? The $p$-adic integres are the disjoint union of $p$ copies of itself scaled by a factor of norm $\frac 1p$, hence the dimension should be $1$. – 2017-02-09
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0Hi Hagen. Yes, I meant the $p$-adic norm; I thought the dimension should be 1, but wanted some confirmation. Thanks. – 2017-02-09
1 Answers
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This question has been answered by Hagen von Eitzen in the comments:
The $p$-adic integers are the disjoint union of $p$ copies of itself scaled by a factor of norm $1/p$, hence the dimension should be $1$.