Abstract
Recent developments in the theory of computing give a canonical way of assigning a dimension to each point of n-dimensional Euclidean space. Computable points have dimension 0, random points have dimension n, and every real number in [0,n] is the dimension of uncountably many points. If X is a reasonably simple subset of n-dimensional Euclidean space (a union of computably closed sets), then the classical Hausdorff dimension of X is just the supremum of the dimensions of the points in X. In this talk I will discuss the meaning of these developments, their implications for both the theory of computing and fractal geometry, and directions for future research.
This research was supported in part by National Science Foundation Grant 0344187.
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Lutz, J.H. (2005). The Dimension of a Point: Computability Meets Fractal Geometry. In: Cooper, S.B., Löwe, B., Torenvliet, L. (eds) New Computational Paradigms. CiE 2005. Lecture Notes in Computer Science, vol 3526. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11494645_37
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DOI: https://doi.org/10.1007/11494645_37
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