31 research outputs found
Lines Missing Every Random Point
We prove that there is, in every direction in Euclidean space, a line that
misses every computably random point. We also prove that there exist, in every
direction in Euclidean space, arbitrarily long line segments missing every
double exponential time random point.Comment: Added a section: "Betting in Doubly Exponential Time.
Dimension Spectra of Lines
This paper investigates the algorithmic dimension spectra of lines in the
Euclidean plane. Given any line L with slope a and vertical intercept b, the
dimension spectrum sp(L) is the set of all effective Hausdorff dimensions of
individual points on L. We draw on Kolmogorov complexity and geometrical
arguments to show that if the effective Hausdorff dimension dim(a, b) is equal
to the effective packing dimension Dim(a, b), then sp(L) contains a unit
interval. We also show that, if the dimension dim(a, b) is at least one, then
sp(L) is infinite. Together with previous work, this implies that the dimension
spectrum of any line is infinite
A Divergence Formula for Randomness and Dimension (Short Version)
If is an infinite sequence over a finite alphabet and is
a probability measure on , then the {\it dimension} of with
respect to , written , is a constructive version of
Billingsley dimension that coincides with the (constructive Hausdorff)
dimension when is the uniform probability measure. This paper
shows that and its dual \Dim^\beta(S), the {\it strong
dimension} of with respect to , can be used in conjunction with
randomness to measure the similarity of two probability measures and
on . Specifically, we prove that the {\it divergence formula}
\dim^\beta(R) = \Dim^\beta(R) =\CH(\alpha) / (\CH(\alpha) + \D(\alpha ||
\beta)) holds whenever and are computable, positive
probability measures on and is random with
respect to . In this formula, \CH(\alpha) is the Shannon entropy of
, and \D(\alpha||\beta) is the Kullback-Leibler divergence between
and
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)
The descriptive theory of represented spaces
This is a survey on the ongoing development of a descriptive theory of
represented spaces, which is intended as an extension of both classical and
effective descriptive set theory to deal with both sets and functions between
represented spaces. Most material is from work-in-progress, and thus there may
be a stronger focus on projects involving the author than an objective survey
would merit.Comment: survey of work-in-progres
Mutual Dimension
We define the lower and upper mutual dimensions and
between any two points and in Euclidean space. Intuitively these are
the lower and upper densities of the algorithmic information shared by and
. We show that these quantities satisfy the main desiderata for a
satisfactory measure of mutual algorithmic information. Our main theorem, the
data processing inequality for mutual dimension, says that, if is computable and Lipschitz, then the inequalities
and hold for all and . We use this inequality and related
inequalities that we prove in like fashion to establish conditions under which
various classes of computable functions on Euclidean space preserve or
otherwise transform mutual dimensions between points.Comment: This article is 29 pages and has been submitted to ACM Transactions
on Computation Theory. A preliminary version of part of this material was
reported at the 2013 Symposium on Theoretical Aspects of Computer Science in
Kiel, German