Translating the Cantor set by a random Randall Dougherty Jack Lutz - - PowerPoint PPT Presentation

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Translating the Cantor set by a random

Randall Dougherty Jack Lutz

• R. Daniel Mauldin

Jason Teutsch arXiv:1205.4821 [cs.CC]

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Goal: Determine how much of the randomness in a “random” real can be cancelled by adding (or subtracting) a member of the Cantor set. Further, determine the range of effective dimensions of points in a random translate of the Cantor set.

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Let E ⊆ Rn. The diameter of E, denoted |E|, is the maximum distance between any two points in E. We will use card for cardinality. A cover G for a set E is a collection

• f sets whose union contains E, and G is a δ-mesh cover if the diameter of each

member G is at most δ. For a number β ≥ 0, the β-dimensional Hausdorff measure of E, written Hβ(E), is given by limδ→0 Hβ

δ (E) where

Hβ

δ (E) = inf

• G∈G

|G|β : G is a countable δ-mesh cover of E

• .

(1) The Hausdorff dimension of a set E, denoted dimH(E), is the unique number α where the α-dimensional Hausdorff measure of E transitions from being negligible to being infinitely large; if β < α, then Hβ(E) = ∞ and if β > α, then Hβ(E) = 0.

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Let Sδ(E) denote the smallest number of sets of diameter at most δ which can cover

• E. The upper box-counting dimension of E is defined as

dimB(E) = lim sup

δ→0

log Sδ(E) − log δ . For all E we have dimH(E) ≤ dimB(E).

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dimH(A × B) ≥ dimH A + dimH B dimH(A × B) ≤ dimH A + dimB B

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The effective (or constructive) β-dimensional Hausdorff measure of a set E, cHβ(Ek), is defined exactly in the same way as Hausdorff measure with the restriction that the covers be uniformly c.e. open sets. This yields the corresponding notion of the effective (or constructive) Hausdorff dimension of a set E, cdimH E.

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We define the constructive dimension of a point x to be the effective Hausdorff dimension of the singleton {x}. Lutz showed that, for any E, cdimH E = sup{cdimH{x} : x ∈ E}. Let E≤α = {x : cdimH{x} ≤ α}. From the above, we know that the effective Hausdorff dimension of E≤α satisfies cdimH E≤α = α; it turns out (Lutz) that also dimH E≤α = α.

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We define the constructive dimension of a point x to be the effective Hausdorff dimension of the singleton {x}. Lutz showed that, for any E, cdimH E = sup{cdimH{x} : x ∈ E}. Let E≤α = {x : cdimH{x} ≤ α}. From the above, we know that the effective Hausdorff dimension of E≤α satisfies cdimH E≤α = α; it turns out (Lutz) that also dimH E≤α = α.

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We define the constructive dimension of a point x to be the effective Hausdorff dimension of the singleton {x}. Lutz showed that, for any E, cdimH E = sup{cdimH{x} : x ∈ E}. Let E≤α = {x : cdimH{x} ≤ α}. From the above, we know that the effective Hausdorff dimension of E≤α satisfies cdimH E≤α = α; it turns out (Lutz) that also dimH E≤α = α.

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The Kolmogorov complexity of a string σ, denoted K(σ), is the length (here we will measure length in ternary units) of the shortest program (under a fixed universal machine) which outputs σ. For a real number x, x ↾ n denotes the first n digits in a ternary expansion of x.

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All sequences s of length n have K(s) ≤ n + O(log n); most of them have K(s) ≥ n − O(1).

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From work of Levin (≥) and Mayordomo (≤) we have for any real number x, cdimH{x} = lim inf

n→∞

K(x ↾ n) n . (2)

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We say a number is Martin-L¨

• f random if it “passes” all Martin-L¨
• f tests. A

Martin-L¨

• f test is a uniformly computably enumerable (c.e.) sequence of open sets

{Um}m∈N with λ(Um) ≤ 2−m, where λ denotes Lebesgue measure. A number x passes such a test if x ∈ ∩mUm. Martin-L¨

• f random reals have high initial segment complexity; indeed every

Martin-L¨

• f random real r satisfies limn K(r ↾ n)/n = 1.

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We say a number is Martin-L¨

• f random if it “passes” all Martin-L¨
• f tests. A

Martin-L¨

• f test is a uniformly computably enumerable (c.e.) sequence of open sets

{Um}m∈N with λ(Um) ≤ 2−m, where λ denotes Lebesgue measure. A number x passes such a test if x ∈ ∩mUm. Martin-L¨

• f random reals have high initial segment complexity; indeed every

Martin-L¨

• f random real r satisfies limn K(r ↾ n)/n = 1.

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Cantor set C ⊆ [0, 1]: dimH C = log3 2 ≈ .6309

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C + C = [0, 2] 1 2C + 1 2C = [0, 1]

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E + C = [0, 2] 1 2E + 1 2C = [0, 1] 1 2E = {00, 02, 11}∞ cdimH E = 1/2

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E + C = [0, 2] 1 2E + 1 2C = [0, 1] 1 2E = {00, 02, 11}∞ cdimH E = 1/2

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E + C = [0, 2] 1 2E + 1 2C = [0, 1] 1 2E = {00, 02, 11}∞ cdimH E = 1/2

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B1 = {0, 1} B2 = {00, 02, 11} B3 = {000, 002, 021, 110, 112} B4 = {0000, 0002, 0011, 0200, 0202, 0211, 1100, 1102, 1111} B5 = {00000, 00002, 00021, 00112, 00210, 01221, 02012, 02110, 02201, 10212, 11010, 11101, 11120, 11122}

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Lemma 1 (Lorentz). There exists a constant c such that for any integer k, if A ⊆ [0, k) is a set of integers with |A| ≥ ℓ ≥ 2, then there exists a set of integers B ⊆ (−k, k) such that A + B ⊇ [0, k) with |B| ≤ ck log ℓ

ℓ .

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So we get E of constructive dimension 1 − dim C such that E + C = [0, 2].

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Theorem 2. Let 1 − dimH C ≤ α ≤ 1 and let r ∈ [0, 1]. Then dimH [(C + r) ∩ E≤α] ≥ α − 1 + dimH C.

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Assume 1 − dimH C < α < 1. Split N into A ⊆ N and ¯ A; then we can write C = CA + C ¯

• A. Choose A of a suitable density D = (1 − α)/dimH C so that dimH CA

comes out to be 1 − α. Then find a closed set E such that cdimH E ≤ α and CA + E = [0, 2]. Let F = 2 − E, so that F − CA = [0, 2]. Let S = C ∩ (F − r); it will suffice to show that dimH S ≥ α − 1 + dimH C. Now for each z ∈ C there exist unique points v ∈ CA and w ∈ C ¯

A such that v + w = z;

let p be the projection map which takes z ∈ C to its unique counterpart w ∈ C ¯

• A. For

each y ∈ C ¯

A we have r + y ∈ [0, 2] ⊆ F − CA, so there exists x ∈ CA such that

r + y ∈ F − x, which gives x + y ∈ S since CA + C ¯

A = C. Thus p maps S onto C ¯ A.

Since p is Lipschitz we have dimH S ≥ dimH C ¯

A ≥ α − 1 + dimH C

because Lipschitz maps do not increase dimension. The theorem follows.

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The set E is constructed as before, but the computation of cdimH E has an additional complication — we have not assumed that α is computable. It turns out that Kolmogorov complexity methods are useful here. Specifically, let A = {⌊y/D⌋ : y ∈ N} . Then initial segments of A are easy to describe: K(A[n]) ≤ 4 log3 n + O(1). Now a straightforward computation shows that, if ǫ > 0, then for all x ∈ E and all sufficiently large k we have K(x ↾ mk) ≤ mk[α + ǫ + o(1)], which is enough to give cdimH E ≤ α.

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Theorem 3. Let 1 − dimH C ≤ α ≤ 1. For every Martin-L¨

• f random real r,

dimH [(C + r) ∩ E≤α] ≤ α − 1 + dimH C.

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Theorem 4. Let 1 − dimH C ≤ α ≤ 1 and let r ∈ [0, 1] be Martin-L¨

• f random. Then

dimH [(C + r) ∩ E=α] = α − 1 + dimH C. Moreover, Hα−1+dimH C [(C + r) ∩ E=α] > 0.

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Questions: How much can the randomness of r be reduced by adding a Cantor set point if r was not completely random to begin with? What about sets other than the Cantor set?