Persistent homology and the upper box dimension, Benjamin Schweinhart (4/3/18)

We prove the first results relating persistent homology to a classically defined fractal dimension. Several previous studies have demonstrated an empirical relationship between persistent homology and fractal dimension; our results are the first rigorous analogue of those comparisons. Specifically, we define a family persistent homology dimensions for a metric space, and exhibit hypotheses under which they are comparable to the upper box dimension. In particular, the dimensions coincide for subsets of R^2 whose upper box dimension exceeds 1.5. This work also raises interesting questions in extremal combinatorics and geometry.