2020
DOI: 10.1007/978-3-030-41672-0_4
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Who Asked Us? How the Theory of Computing Answers Questions about Analysis

Abstract: Algorithmic fractal dimensions-constructs of computability theory-have recently been used to answer open questions in classical geometric measure theory, questions of mathematical analysis whose statements do not involve computability theory or logic. We survey these developments and the prospects for future such results.

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Cited by 7 publications
(1 citation statement)
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“…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 [27], 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 randomness [35]) and partly because of its applications to classical fractal geometry [30,31]. Algorithmic dimension plays a motivating role in this paper, but resource-bounded dimension is our main topic.…”
Section: Introductionmentioning
confidence: 99%
“…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 [27], 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 randomness [35]) and partly because of its applications to classical fractal geometry [30,31]. Algorithmic dimension plays a motivating role in this paper, but resource-bounded dimension is our main topic.…”
Section: Introductionmentioning
confidence: 99%