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arXiv · 1102.5136

Hausdorff dimension for fractals invariant under the multiplicative integers

Richard Kenyon·Yuval Peres·Boris Solomyak
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Abstract

We consider subsets of the (symbolic) sequence space that are invariant under the action of the semigroup of multiplicative integers. A representative example is the collection of all 0-1 sequences $(x_k)$ such that $x_k x_{2k}=0$ for all $k$. We compute the Hausdorff and Minkowski dimensions of these sets and show that they are typically different. The proof proceeds via a variational principle for multiplicative subshifts.

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Richard Kenyon, Yuval Peres, Boris Solomyak. 2011-02-25. Hausdorff dimension for fractals invariant under the multiplicative integers. https://arxiv.org/abs/1102.5136

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