1,168 research outputs found
Fractal Intersections and Products via Algorithmic Dimension
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
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
The point-to-set principle characterizes the Hausdorff dimension of a subset
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 . One of the primary
motivations of this definition is that, if has optimal oracles, then the
conclusion of Marstrand's projection theorem holds for . We show that every
analytic set has optimal oracles. We also prove that if the Hausdorff and
packing dimensions of agree, then 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
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
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 . 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 . We first extend two
fractal dimensions--computability-theoretic versions of classical Hausdorff and
packing dimensions that assign dimensions and to
individual points --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 used to
construct a self-similar fractal in the plane are elements of the
hyperspace of the plane, and they converge to 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 that are analytic, i.e.,
, a tight bound on the packing dimension of the hyperspace
of in terms of the packing dimension of itself
The status and programs of the New Relativity Theory
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 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
-brane propagator in dimensions ( [18]) to the full
multidimensional case involving all -branes : the construction of the
Multidimensional-Particle propagator in Clifford spaces (-spaces) associated
with a nested family of -loop histories living in a target -dim
background spacetime . We show how the effective -space geometry is related
to curvature of ordinary spacetime. The motion of rigid
particles/branes is studied to explain the natural of classical
spin. The relation among -space geometry and , Finsler Geometry
and (Braided) Quantum Groups is discussed. Some final remarks about the
Riemannian long distance limit of -space geometry are made.Comment: Tex file, 21 page
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