What is it about?
Fractals are shapes that have intricate structure at every scale. We design an optimal yardstick for fractal sets. Using it, we unify two seemingly different views of intricacy, point by point: geometric density nearby and the difficulty of specifying that location.
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Why is it important?
Fractals appear across science and technology, from natural patterns to modern datasets. Our results provide a unified view linking geometry and information, opening the door to simpler arguments and sharper results in the study of complex sets.
Perspectives
As a grad student, I defined a specific measure and noticed that it made two ways of thinking about local complexity line up. After I graduated and moved back to Iowa, my dad and I started asking, what made that measure so special? Pursuing that question led to our optimality definitions and results.
Neil Lutz
Swarthmore College
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This page is a summary of: Algorithmically Optimal Outer Measures, ACM Transactions on Computation Theory, November 2025, ACM (Association for Computing Machinery),
DOI: 10.1145/3733607.
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