Persistent Homology and the Upper Box Dimension. (arXiv:1802.00533v1 [math.MG])

We introduce a fractal dimension for a metric space based on the persistent homology of subsets of that space. We exhibit hypotheses under which this dimension is comparable to the upper box dimension; in particular, the dimensions coincide for subsets of $\mathbb{R}^2$ whose upper box dimension exceeds $1.5.$

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