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235
Dimension in Complexity Classes
 SIAM Journal on Computing
, 2000
"... A theory of resourcebounded dimension is developed using gales, which are natural generalizations of martingales. When the resource bound (a parameter of the theory) is unrestricted, the resulting dimension is precisely the classical Hausdorff dimension (sometimes called "fractal dimension"). Othe ..."
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Cited by 115 (17 self)
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A theory of resourcebounded dimension is developed using gales, which are natural generalizations of martingales. When the resource bound (a parameter of the theory) is unrestricted, the resulting dimension is precisely the classical Hausdorff dimension (sometimes called "fractal dimension"). Other choices of the parameter yield internal dimension theories in E, E 2 , ESPACE, and other complexity classes, and in the class of all decidable problems. In general, if C is such a class, then every set X of languages has a dimension in C, which is a real number dim(X j C) 2 [0; 1]. Along with the elements of this theory, two preliminary applications are presented: 1. For every real number 0 1 2 , the set FREQ( ), consisting of all languages that asymptotically contain at most of all strings, has dimension H()  the binary entropy of  in E and in E 2 . 2. For every real number 0 1, the set SIZE( 2 n n ), consisting of all languages decidable by Boolean circuits of at most 2 n n gates, has dimension in ESPACE.
The Dimensions of Individual Strings and Sequences
 INFORMATION AND COMPUTATION
, 2003
"... A constructive version of Hausdorff dimension is developed using constructive supergales, which are betting strategies that generalize the constructive supermartingales used in the theory of individual random sequences. This constructive dimension is used to assign every individual (infinite, binary ..."
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Cited by 95 (10 self)
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A constructive version of Hausdorff dimension is developed using constructive supergales, which are betting strategies that generalize the constructive supermartingales used in the theory of individual random sequences. This constructive dimension is used to assign every individual (infinite, binary) sequence S a dimension, which is a real number dim(S) in the interval [0, 1]. Sequences that
Dimensions and measures in infinite iterated function systems
 PROC. LONDON MATH. SOC
, 1996
"... The Hausdorff and packing measures and dimensions of the limit sets of iterated function systems generated by countable families of conformal contractions are investigated. Conformal measures for such systems, reflecting geometric properties of the limit set, are introduced, proven to exist, and to ..."
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Cited by 82 (23 self)
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The Hausdorff and packing measures and dimensions of the limit sets of iterated function systems generated by countable families of conformal contractions are investigated. Conformal measures for such systems, reflecting geometric properties of the limit set, are introduced, proven to exist, and to be unique. The existence of a unique invariant probability equivalent to the conformal measure is derived. Our methods employ the concepts of the PerronFrobenius operator, symbolic dynamics on an infinite dimensional shift space, and the properties of the above mentioned ergodic invariant measure. A formula for the Hausdorff dimension of the limit set in terms of the pressure function is derived. Fractal phenomena not exhibited by finite systems are shown to appear in the infinite case. In particular a variety of conditions are provided for Hausdorff and packing measures to be positive or finite, and a number of examples are described showing the appearance of various possible combinations for these quantities. One example given special attention is the limit set associated to the complex continued fraction expansion  in particular lower and upper estimates for its Hausdor dimension are given. A large natural class of systems whose limit sets are "dimensionless in the restricted sense" is described.
Recent work connected with the Kakeya problem
 Prospects in mathematics (Princeton, NJ
, 1996
"... A Kakeya set in R n is a compact set E ⊂ R n containing a unit line segment in every direction, i.e. ∀e ∈ S n−1 ∃x ∈ R n: x + te ∈ E ∀t ∈ [ − 1 1 ..."
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Cited by 62 (2 self)
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A Kakeya set in R n is a compact set E ⊂ R n containing a unit line segment in every direction, i.e. ∀e ∈ S n−1 ∃x ∈ R n: x + te ∈ E ∀t ∈ [ − 1 1
FiniteState Dimension
, 2001
"... Classical Hausdorff dimension (sometimes called fractal dimension) was recently effectivized using gales (betting strategies that generalize martingales), thereby endowing various complexity classes with dimension structure and also defining the constructive dimensions of individual binary (infinite ..."
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Cited by 40 (16 self)
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Classical Hausdorff dimension (sometimes called fractal dimension) was recently effectivized using gales (betting strategies that generalize martingales), thereby endowing various complexity classes with dimension structure and also defining the constructive dimensions of individual binary (infinite) sequences. In this paper we use gales computed by multiaccount finitestate gamblers to develop the finitestate dimensions of sets of binary sequences and individual binary sequences. The theorem of Eggleston (1949) relating Hausdorff dimension to entropy is shown to hold for finitestate dimension, both in the space of all sequences and in the space of all rational sequences (binary expansions of rational numbers). Every rational sequence has finitestate dimension 0, but every rational number in [0; 1] is the finitestate dimension of a sequence in the lowlevel complexity class AC0 . Our main theorem shows that the finitestate dimension of a sequence is precisely the infimum of all compression ratios achievable on the sequence by informationlossless finitestate compressors.
Mellin Transforms And Asymptotics: Digital Sums
, 1993
"... Arithmetic functions related to number representation systems exhibit various periodicity phenomena. For instance, a well known theorem of Delange expresses the total number of ones in the binary representations of the first n integers in terms of a periodic fractal function. We show that such perio ..."
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Cited by 39 (13 self)
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Arithmetic functions related to number representation systems exhibit various periodicity phenomena. For instance, a well known theorem of Delange expresses the total number of ones in the binary representations of the first n integers in terms of a periodic fractal function. We show that such periodicity phenomena can be analyzed rather systematically using classical tools from analytic number theory, namely the MellinPerron formulae. This approach yields naturally the Fourier series involved in the expansions of a variety of digital sums related to number representation systems.
A generalization of Chaitin’s halting probability Ω and halting selfsimilar sets
 Hokkaido Math. J
, 2002
"... We generalize the concept of randomness in an infinite binary sequence in order to characterize the degree of randomness by a real number D> 0. Chaitin’s halting probability Ω is generalized to Ω D whose degree of randomness is precisely D. On the basis of this generalization, we consider the degree ..."
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Cited by 34 (12 self)
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We generalize the concept of randomness in an infinite binary sequence in order to characterize the degree of randomness by a real number D> 0. Chaitin’s halting probability Ω is generalized to Ω D whose degree of randomness is precisely D. On the basis of this generalization, we consider the degree of randomness of each point in Euclidean space through its basetwo expansion. It is then shown that the maximum value of such a degree of randomness provides the Hausdorff dimension of a selfsimilar set that is computable in a certain sense. The class of such selfsimilar sets includes familiar fractal sets such as the Cantor set, von Koch curve, and Sierpiński gasket. Knowledge of the property of Ω D allows us to show that the selfsimilar subset of [0,1] defined by the halting set of a universal algorithm has a Hausdorff dimension of one.
Shape Dimension and Approximation from Samples
, 2003
"... There are many scientific and engineering applications where an automatic detection of shape dimension from sample data is necessary. Topological dimensions of shapes constitute an important global feature of them. We present a Voronoi based dimension detection algorithm that assigns a dimension to ..."
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Cited by 31 (6 self)
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There are many scientific and engineering applications where an automatic detection of shape dimension from sample data is necessary. Topological dimensions of shapes constitute an important global feature of them. We present a Voronoi based dimension detection algorithm that assigns a dimension to a sample point which is the topological dimension of the manifold it belongs to. Based on this dimension detection, we also present an algorithm to approximate shapes of arbitrary dimension from their samples. Our empirical results with data sets in three dimensions support our theory.
Multidimensional Van Der Corput And Sublevel Set Estimates
 J. Amer. Math. Soc
"... This paper is devoted to quantifying that principle for functions of several variables, particularly as it pertains to two problems in harmonic analysis. Along the way we shall encounter diverse problems and techniques, and shall be led to issues distinctly combinatorial in nature. We begin by revie ..."
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Cited by 31 (4 self)
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This paper is devoted to quantifying that principle for functions of several variables, particularly as it pertains to two problems in harmonic analysis. Along the way we shall encounter diverse problems and techniques, and shall be led to issues distinctly combinatorial in nature. We begin by reviewing two wellknown questions in onedimensional analysis. Suppose that u is a (smooth) real valued function on the real line R such that for some