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Algorithmic randomness, reverse mathematics, and the dominated convergence theorem
 Ann. Pure Appl. Logic
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Symbolic dynamics: entropy = dimension = complexity
"... Let d be a positive integer. Let G be the additive monoid N d or the additive group Z d. Let A be a finite set of symbols. The shift action of G on A G is given by S g (x)(h) = x(g + h) for all g,h ∈ G and all x ∈ A G. A Gsubshift is defined to be a nonempty closed set X ⊆ A G such that S g (x) ∈ ..."
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Let d be a positive integer. Let G be the additive monoid N d or the additive group Z d. Let A be a finite set of symbols. The shift action of G on A G is given by S g (x)(h) = x(g + h) for all g,h ∈ G and all x ∈ A G. A Gsubshift is defined to be a nonempty closed set X ⊆ A G such that S g (x) ∈ X for all g ∈ G and all x ∈ X. Given a Gsubshift X, the topological entropy ent(X) is defined as usual [31]. The standard metric on A G is defined by ρ(x,y) = 2 −Fn  where n is as large as possible such that x↾Fn = y↾Fn. Here Fn = {0,1,...,n} d if G = N d, and Fn = {−n,...,−1,0,1,...,n} d if G = Z d. For any X ⊆ A G the Hausdorff dimension dim(X) and the effective Hausdorff dimension effdim(X) are defined as usual [14, 26, 27] with respect to the standard metric. It is well known that effdim(X) = sup x∈X liminfnK(x↾Fn)/Fn  where K denotes Kolmogorov complexity [9]. If X is a Gsubshift, we prove that ent(X) = dim(X) = effdim(X), and ent(X) ≥ limsup n K(x↾Fn)/Fn  for all x ∈ X, and ent(X) = limnK(x↾Fn)/Fn  for some x ∈ X.
Annals of Pure and Applied Logic
"... This article appeared in a journal published by Elsevier. The attached copy is furnished to the author for internal noncommercial research and education use, including for instruction at the authors institution and sharing with colleagues. Other uses, including reproduction and distribution, or sel ..."
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This article appeared in a journal published by Elsevier. The attached copy is furnished to the author for internal noncommercial research and education use, including for instruction at the authors institution and sharing with colleagues. Other uses, including reproduction and distribution, or selling or licensing copies, or posting to personal, institutional or third party websites are prohibited. In most cases authors are permitted to post their version of the article (e.g. in Word or Tex form) to their personal website or institutional repository. Authors requiring further information regarding Elsevier’s archiving and manuscript policies are encouraged to visit:
CCACCR Conference
, 2011
"... Let G be (N d,+) or (Z d,+) where d ≥ 1. Let A be a finite set of symbols. We endow A with the discrete topology and A G with the product topology. The shift action of G on A G is given by (S g x)(h) = x(g + h) for g, h ∈ G and x ∈ A G. A subshift is a nonempty set X ⊆ A G which is topologically cl ..."
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Let G be (N d,+) or (Z d,+) where d ≥ 1. Let A be a finite set of symbols. We endow A with the discrete topology and A G with the product topology. The shift action of G on A G is given by (S g x)(h) = x(g + h) for g, h ∈ G and x ∈ A G. A subshift is a nonempty set X ⊆ A G which is topologically closed and shiftinvariant, i.e., x ∈ X implies S g x ∈ X for all g ∈ G. Symbolic dynamics is the study of subshifts. If X ⊆ A G and Y ⊆ B G are subshifts, a shift morphism from X to Y is a continuous mapping Φ: X → Y such that Φ(S g x) = S g Φ(x) for all x ∈ X and g ∈ G. By compactness, any shift morphism Φ is given by a block code, i.e., a finite mapping φ: A F → B where F is a finite subset of G and Φ(x)(g) = φ(S g x↾F) for all x ∈ X and g ∈ G. 2 Some new (!?!) results on subshifts: Let d be a positive integer, let A be a finite set of symbols, and let X be a nonempty subset of A G where G is N d or Z d. The Hausdorff dimension, dim(X), and the effective Hausdorff dimension, effdim(X), are defined as usual with respect to the standard metric ρ(x, y) = 2 −Fn where n is as large as possible such that x↾Fn = y↾Fn. Here Fn is {1,..., n} d if G = N d, or {−n,..., n} d if G = Z d. We first state some old results. 1. effdim(X) = sup x∈X effdim(x). 2. effdim(x) = liminf n→∞
Finding Subsets of Positive Measure∗
, 2014
"... An important theorem of geometric measure theory (first proved by Besicovitch and Davies for Euclidean space) says that every analytic set of nonzero sdimensional Hausdorff measure Hs contains a closed subset of nonzero (and indeed finite) Hsmeasure. We investigate the question how hard it is ..."
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An important theorem of geometric measure theory (first proved by Besicovitch and Davies for Euclidean space) says that every analytic set of nonzero sdimensional Hausdorff measure Hs contains a closed subset of nonzero (and indeed finite) Hsmeasure. We investigate the question how hard it is to find such a set, in terms of the index set complexity, and in terms of the complexity of the parameter needed to define such a closed set. Among other results, we show that given a (lightface) Σ11 set of reals in Cantor space, there is always a Π 0 1(O) subset on nonzero Hsmeasure definable from Kleene’s O. On the other hand, there are Π02 sets of reals where no hyperarithmetic real can define a closed subset of nonzero measure. 1