Algorithmic Fractal Dimensions in Geometric Measure Theory

2023 EUROPEAN SUMMER MEETING OF THE ASSOCIATION FOR SYMBOLIC LOGIC LOGIC COLLOQUIUM 2023 University of Milan Milan, Italy June 5 – June 9, 2023

“…The point-to-set principle (PSP) of J. Lutz and N. Lutz [8] fully characterizes Hausdorff and packing dimensions in terms of effective dimensions in the Euclidean space, enabling effective dimensions to be used to answer open questions about fractal geometry, with already an interesting list of geometric measure theory results (see [3,11] and more recent results in [7,14,16,15]).…”

Dimension and the Structure of Complexity Classes

“…For example this theory defines, for every subset X of C, a quasipolynomial-time (i.e., n polylog n -time) dimension dim qp (X) in such a way that dim(X | EXP) = dim qp (X ∩ EXP) is a coherent notion of the dimension of X within the complexity class EXP = TIME(2 polynomial ). The second method [26], algorithmic dimension (also called constructive dimension or effective dimension) has to date been more widely investigated, partly because of its interactions with algorithmic randomness (i.e., Martin-Löf ran-domness [35]) and partly because of its applications to classical fractal geometry [29,31]. Algorithmic dimension plays a motivating role in this paper, but resource-bounded dimension is our main topic.…”

2023 NORTH AMERICAN ANNUAL MEETING OF THE ASSOCIATION FOR SYMBOLIC LOGIC University of California, Irvine Irvine, CA, USA March 25–29, 2023

“…The point-to-set principle of Lutz and Lutz [8] fully characterizes Hausdorff and packing dimensions in terms of effective dimensions in the Euclidean space, enabling effective dimensions to be used to answer open questions about fractal geometry, with already an interesting list of geometric measure theory results (see [3,11] and more recent results in [7,14,15,16]).…”