1,168 research outputs found

    Fractal Intersections and Products via Algorithmic Dimension

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    Algorithmic dimensions quantify the algorithmic information density of individual points and may be defined in terms of Kolmogorov complexity. This work uses these dimensions to bound the classical Hausdorff and packing dimensions of intersections and Cartesian products of fractals in Euclidean spaces. This approach shows that a known intersection formula for Borel sets holds for arbitrary sets, and it significantly simplifies the proof of a known product formula. Both of these formulas are prominent, fundamental results in fractal geometry that are taught in typical undergraduate courses on the subject

    The Point-to-Set Principle, the Continuum Hypothesis, and the Dimensions of Hamel Bases

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    We prove that the Continuum Hypothesis implies that every real number in (0,1] is the Hausdorff dimension of a Hamel basis of the vector space of reals over the field of rationals. The logic of our proof is of particular interest. The statement of our theorem is classical; it does not involve the theory of computing. However, our proof makes essential use of algorithmic fractal dimension--a computability-theoretic construct--and the point-to-set principle of J. Lutz and N. Lutz (2018)

    Optimal Oracles for Point-To-Set Principles

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    The point-to-set principle characterizes the Hausdorff dimension of a subset ERnE\subseteq\R^n by the effective dimension of its individual points. This characterization has been used to prove several results in classical, i.e., without any computability requirements, analysis. Recent work has shown that algorithmic techniques can be fruitfully applied to Marstrand's projection theorem, a fundamental result in fractal geometry. In this paper, we introduce an extension of point-to-set principle - the notion of optimal oracles for subsets ERnE\subseteq\R^n. One of the primary motivations of this definition is that, if EE has optimal oracles, then the conclusion of Marstrand's projection theorem holds for EE. We show that every analytic set has optimal oracles. We also prove that if the Hausdorff and packing dimensions of EE agree, then EE has optimal oracles. Thus, the existence of optimal oracles subsume the currently known sufficient conditions for Marstrand's theorem to hold. Under certain assumptions, every set has optimal oracles. However, assuming the axiom of choice and the continuum hypothesis, we construct sets which do not have optimal oracles. This construction naturally leads to a new, algorithmic, proof of Davies theorem on projections

    Projection Theorems Using Effective Dimension

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    In this paper we use the theory of computing to study fractal dimensions of projections in Euclidean spaces. A fundamental result in fractal geometry is Marstrand\u27s projection theorem, which shows that for every analytic set E, for almost every line L, the Hausdorff dimension of the orthogonal projection of E onto L is maximal. We use Kolmogorov complexity to give two new results on the Hausdorff and packing dimensions of orthogonal projections onto lines. The first shows that the conclusion of Marstrand\u27s theorem holds whenever the Hausdorff and packing dimensions agree on the set E, even if E is not analytic. Our second result gives a lower bound on the packing dimension of projections of arbitrary sets. Finally, we give a new proof of Marstrand\u27s theorem using the theory of computing

    Extending the Reach of the Point-To-Set Principle

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    The point-to-set principle of J. Lutz and N. Lutz (2018) has recently enabled the theory of computing to be used to answer open questions about fractal geometry in Euclidean spaces Rn\mathbb{R}^n. These are classical questions, meaning that their statements do not involve computation or related aspects of logic. In this paper we extend the reach of the point-to-set principle from Euclidean spaces to arbitrary separable metric spaces XX. We first extend two fractal dimensions--computability-theoretic versions of classical Hausdorff and packing dimensions that assign dimensions dim(x)\dim(x) and Dim(x)\textrm{Dim}(x) to individual points xXx\in X--to arbitrary separable metric spaces and to arbitrary gauge families. Our first two main results then extend the point-to-set principle to arbitrary separable metric spaces and to a large class of gauge families. We demonstrate the power of our extended point-to-set principle by using it to prove new theorems about classical fractal dimensions in hyperspaces. (For a concrete computational example, the stages E0,E1,E2,E_0, E_1, E_2, \ldots used to construct a self-similar fractal EE in the plane are elements of the hyperspace of the plane, and they converge to EE in the hyperspace.) Our third main result, proven via our extended point-to-set principle, states that, under a wide variety of gauge families, the classical packing dimension agrees with the classical upper Minkowski dimension on all hyperspaces of compact sets. We use this theorem to give, for all sets EE that are analytic, i.e., Σ11\mathbf{\Sigma}^1_1, a tight bound on the packing dimension of the hyperspace of EE in terms of the packing dimension of EE itself

    The status and programs of the New Relativity Theory

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    A review of the most recent results of the New Relativity Theory is presented. These include a straightforward derivation of the Black Hole Entropy-Area relation and its logarithmiclogarithmic corrections; the derivation of the string uncertainty relations and generalizations ; ; the relation between the four dimensional gravitational conformal anomaly and the fine structure constant; the role of Noncommutative Geometry, Negative Probabilities and Cantorian-Fractal spacetime in the Young's two-slit experiment. We then generalize the recent construction of the Quenched-Minisuperspace bosonic pp-brane propagator in DD dimensions (AACSAACS [18]) to the full multidimensional case involving all pp-branes : the construction of the Multidimensional-Particle propagator in Clifford spaces (CC-spaces) associated with a nested family of pp-loop histories living in a target DD-dim background spacetime . We show how the effective CC-space geometry is related to extrinsicextrinsic curvature of ordinary spacetime. The motion of rigid particles/branes is studied to explain the natural emergenceemergence of classical spin. The relation among CC-space geometry and W{\cal W}, Finsler Geometry and (Braided) Quantum Groups is discussed. Some final remarks about the Riemannian long distance limit of CC-space geometry are made.Comment: Tex file, 21 page
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