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11
Infinite subsets of random sets of integers
 Math. Res. Lett
, 2009
"... Abstract. There is an infinite subset of a MartinLöf random set of integers that does not compute any MartinLöf random set of integers. To prove this, we show that each real of positive effective Hausdorff dimension computes an infinite subset of a MartinLöf random set of integers, and apply a re ..."
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Abstract. There is an infinite subset of a MartinLöf random set of integers that does not compute any MartinLöf random set of integers. To prove this, we show that each real of positive effective Hausdorff dimension computes an infinite subset of a MartinLöf random set of integers, and apply a result of Miller. 1.
Ktrivial closed sets and continuous functions
 in Proceedings of CIE 2007
"... Abstract. We investigate the notion of Ktriviality for closed sets and continuous functions. Every Ktrivial closed set contains a Ktrivial real. There exists a Ktrivial Π 0 1 class with no computable elements. For any Ktrivial degree d, there is a Ktrivial continuous function of degree d. 1 ..."
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Abstract. We investigate the notion of Ktriviality for closed sets and continuous functions. Every Ktrivial closed set contains a Ktrivial real. There exists a Ktrivial Π 0 1 class with no computable elements. For any Ktrivial degree d, there is a Ktrivial continuous function of degree d. 1
ALGORITHMIC RANDOMNESS AND CAPACITY OF CLOSED SETS
"... Abstract. We investigate the connection between measure, capacity and algorithmic randomness for the space of closed sets. For any computable measure m, a computable capacity T may be defined by letting T (Q) be the measure of the family of closed sets K which have nonempty intersection with Q. We p ..."
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Abstract. We investigate the connection between measure, capacity and algorithmic randomness for the space of closed sets. For any computable measure m, a computable capacity T may be defined by letting T (Q) be the measure of the family of closed sets K which have nonempty intersection with Q. We prove an effective version of Choquet’s capacity theorem by showing that every computable capacity may be obtained from a computable measure in this way. We establish conditions on the measure m that characterize when the capacity of an mrandom closed set equals zero. This includes new results in classical probability theory as well as results for algorithmic randomness. For certain computable measures, we construct effectively closed sets with positive capacity and with Lebesgue measure zero. We show that for computable measures, a real q is upper semicomputable if and only if there is an effectively closed set with capacity q.
Algorithmic Randomness of Continuous Functions, in preparation
"... We investigate notions of randomness in the space C(2 N) of continuous functions on 2 N. A probability measure is given and a version of the MartinLöf Test for randomness is defined. Random ∆ 0 2 continuous functions exist, but no computable function can be random and no random function can map a c ..."
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We investigate notions of randomness in the space C(2 N) of continuous functions on 2 N. A probability measure is given and a version of the MartinLöf Test for randomness is defined. Random ∆ 0 2 continuous functions exist, but no computable function can be random and no random function can map a computable real to a computable real. The image of a random continuous function is always a perfect set and hence uncountable. For any y ∈ 2 N, there exists a random continuous function F with y in the image of F. Thus the image of a random continuous function need not be a random closed set. The set of zeroes of a random continuous function is always a random closed set.
ALGORITHMICALLY RANDOM CLOSED SETS AND PROBABILITY
"... by Logan M. Axon Algorithmic randomness in the Cantor space, 2 ω, has recently become the subject of intense study. Originally defined in terms of the fair coin measure, algorithmic randomness has since been extended, for example in Reimann and Slaman [22, 23], to more general measures. Others have ..."
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by Logan M. Axon Algorithmic randomness in the Cantor space, 2 ω, has recently become the subject of intense study. Originally defined in terms of the fair coin measure, algorithmic randomness has since been extended, for example in Reimann and Slaman [22, 23], to more general measures. Others have meanwhile developed definitions of algorithmic randomness for different spaces, for example the space of continuous functions on the unit interval (Fouché [8, 9]), more general topological spaces (Hertling and Weihrauch [12]), and the closed subsets of 2 ω (Barmpalias et al. [1], KjosHanssen and Diamondstone [14]). Our work has also been to develop a definition of algorithmically random closed subsets.
Effective Capacity and Randomness of Closed Sets ⋆
, 2010
"... We investigate the connection between measure and capacity for the space C of nonempty closed subsets of 2 N. For any computable measure µ ∗ , a computable capacity T may be defined by letting T (Q) be the measure of the family of closed sets K which have nonempty intersection with Q. We prove an ef ..."
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We investigate the connection between measure and capacity for the space C of nonempty closed subsets of 2 N. For any computable measure µ ∗ , a computable capacity T may be defined by letting T (Q) be the measure of the family of closed sets K which have nonempty intersection with Q. We prove an effective version of the Choquet’s theorem by showing that every computable capacity may be obtained from a computable measure in this way. We establish conditions that characterize when the capacity of a random closed set equals zero or is> 0. We construct for certain measures an effectively closed set with positive capacity and with Lebesgue measure zero.
Immunity for Closed Sets ⋆
"... Abstract. The notion of immune sets is extended to closed sets and Π 0 1 classes in particular. We construct a Π 0 1 class with no computable member which is not immune. We show that for any computably inseparable sets A and B, the class S(A, B) of separating sets for A and B is immune. We show that ..."
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Abstract. The notion of immune sets is extended to closed sets and Π 0 1 classes in particular. We construct a Π 0 1 class with no computable member which is not immune. We show that for any computably inseparable sets A and B, the class S(A, B) of separating sets for A and B is immune. We show that every perfect thin Π 0 1 class is immune. We define the stronger notion of prompt immunity and construct an example of a Π 0 1 class of positive measure which is promptly immune. We show that the immune degrees in the Medvedev lattice of closed sets forms a filter. We show that for any Π 0 1 class P with no computable element, there is a Π 0 1 class Q which is not immune and has no computable element, and which is Medvedev reducible to P. We show that any random closed set is immune.
Dimension spectra of random subfractals of selfsimilar fractals
, 2012
"... The (constructive Hausdorff) dimension of a point x in Euclidean space is the algorithmic information density of x. Roughly speaking, this is the least real number dim(x) such that r×dim(x) bits suffices to specify x on a generalpurpose computer with arbitrarily high precisions 2−r. The dimension s ..."
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The (constructive Hausdorff) dimension of a point x in Euclidean space is the algorithmic information density of x. Roughly speaking, this is the least real number dim(x) such that r×dim(x) bits suffices to specify x on a generalpurpose computer with arbitrarily high precisions 2−r. The dimension spectrum of a set X in Euclidean space is the subset of [0, n] consisting of the dimensions of all points in X. The dimensions of points have been shown to be geometrically meaningful (Lutz 2003, Hitchcock 2003), and the dimensions of points in selfsimilar fractals have been completely analyzed (Lutz and Mayordomo 2008). Here we begin the more challenging task of analyzing the dimensions of points in random fractals. We focus on fractals that are randomly selected subfractals of a given selfsimilar fractal. We formulate the specification of a point in such a subfractal as the outcome of an infinite twoplayer game between a selector that selects the subfractal and a coder that selects a point within the subfractal. Our selectors are algorithmically random with respect to various probability measures, so our selectorcoder games are, from the coder’s point of view, games against nature.
E ective Randomness of Unions and Intersections
, 2012
"... We investigate the Abstractrandomness of unions and intersections of random sets under various notions of randomness corresponding to di erent probability measures. For example, the union of two relatively MartinLof random sets is not MartinLof random but is random with respect to the Bernoulli m ..."
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We investigate the Abstractrandomness of unions and intersections of random sets under various notions of randomness corresponding to di erent probability measures. For example, the union of two relatively MartinLof random sets is not MartinLof random but is random with respect to the Bernoulli measure 3 4 under which any number belongs to the set with probability 3 4. Conversely, any 3 random set is the union of two 4 MartinLof random sets. Unions and intersections of random closed sets are also studied. 1