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

1 Translating the Cantor set by a random Randall Dougherty Jack Lutz R. Daniel Mauldin Jason Teutsch arXiv:1205.4821 [cs.CC]

2 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.

3 Let E ⊆ R n . 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 of 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 H β ( E ) , is given by lim δ → 0 H β δ ( E ) where �� � | G | β : G is a countable δ -mesh cover of E H β δ ( E ) = inf . (1) G ∈G The Hausdorff dimension of a set E , denoted dim H ( 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 H β ( E ) = ∞ and if β > α , then H β ( E ) = 0 .

4 Let S δ ( 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 log S δ ( E ) dim B ( E ) = lim sup − log δ . δ → 0 For all E we have dim H ( E ) ≤ dim B ( E ) .

5 dim H ( A × B ) ≥ dim H A + dim H B dim H ( A × B ) ≤ dim H A + dim B B

6 The effective (or constructive ) β -dimensional Hausdorff measure of a set E , c H β ( E k ) , 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 , cdim H E .

7 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 , cdim H E = sup { cdim H { x } : x ∈ E } . Let E ≤ α = { x : cdim H { x } ≤ α } . From the above, we know that the effective Hausdorff dimension of E ≤ α satisfies cdim H E ≤ α = α ; it turns out (Lutz) that also dim H E ≤ α = α .

8 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 , cdim H E = sup { cdim H { x } : x ∈ E } . Let E ≤ α = { x : cdim H { x } ≤ α } . From the above, we know that the effective Hausdorff dimension of E ≤ α satisfies cdim H E ≤ α = α ; it turns out (Lutz) that also dim H E ≤ α = α .

9 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 , cdim H E = sup { cdim H { x } : x ∈ E } . Let E ≤ α = { x : cdim H { x } ≤ α } . From the above, we know that the effective Hausdorff dimension of E ≤ α satisfies cdim H E ≤ α = α ; it turns out (Lutz) that also dim H E ≤ α = α .

10 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 .

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

12 From work of Levin ( ≥ ) and Mayordomo ( ≤ ) we have for any real number x , K ( x ↾ n ) cdim H { x } = lim inf . (2) n n →∞

13 We say a number is Martin-L¨ of random if it “passes” all Martin-L¨ of tests. A Martin-L¨ of test is a uniformly computably enumerable (c.e.) sequence of open sets { U m } m ∈ N with λ ( U m ) ≤ 2 − m , where λ denotes Lebesgue measure. A number x passes such a test if x �∈ ∩ m U m . Martin-L¨ of random reals have high initial segment complexity; indeed every of random real r satisfies lim n K ( r ↾ n ) /n = 1 . Martin-L¨

14 We say a number is Martin-L¨ of random if it “passes” all Martin-L¨ of tests. A Martin-L¨ of test is a uniformly computably enumerable (c.e.) sequence of open sets { U m } m ∈ N with λ ( U m ) ≤ 2 − m , where λ denotes Lebesgue measure. A number x passes such a test if x �∈ ∩ m U m . Martin-L¨ of random reals have high initial segment complexity; indeed every of random real r satisfies lim n K ( r ↾ n ) /n = 1 . Martin-L¨

15 Cantor set C ⊆ [0 , 1] : dim H C = log 3 2 ≈ . 6309

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

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

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

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

20 B 1 = { 0 , 1 } B 2 = { 00 , 02 , 11 } B 3 = { 000 , 002 , 021 , 110 , 112 } B 4 = { 0000 , 0002 , 0011 , 0200 , 0202 , 0211 , 1100 , 1102 , 1111 } B 5 = { 00000 , 00002 , 00021 , 00112 , 00210 , 01221 , 02012 , 02110 , 02201 , 10212 , 11010 , 11101 , 11120 , 11122 }

21 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 .

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

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

24 Assume 1 − dim H C < α < 1 . Split N into A ⊆ N and ¯ A ; then we can write C = C A + C ¯ A . Choose A of a suitable density D = (1 − α ) / dim H C so that dim H C A comes out to be 1 − α . Then find a closed set E such that cdim H E ≤ α and C A + E = [0 , 2] . Let F = 2 − E , so that F − C A = [0 , 2] . Let S = C ∩ ( F − r ) ; it will suffice to show that dim H S ≥ α − 1 + dim H C . Now for each z ∈ C there exist unique points v ∈ C A 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 − C A , so there exists x ∈ C A such that r + y ∈ F − x , which gives x + y ∈ S since C A + C ¯ A = C . Thus p maps S onto C ¯ A . Since p is Lipschitz we have dim H S ≥ dim H C ¯ A ≥ α − 1 + dim H C because Lipschitz maps do not increase dimension. The theorem follows.

25 The set E is constructed as before, but the computation of cdim H 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 log 3 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 ↾ m k ) ≤ m k [ α + ǫ + o (1)] , which is enough to give cdim H E ≤ α .

26 Theorem 3. Let 1 − dim H C ≤ α ≤ 1 . For every Martin-L¨ of random real r , dim H [( C + r ) ∩ E ≤ α ] ≤ α − 1 + dim H C .

27 Theorem 4. Let 1 − dim H C ≤ α ≤ 1 and let r ∈ [0 , 1] be Martin-L¨ of random. Then dim H [( C + r ) ∩ E = α ] = α − 1 + dim H C . Moreover, H α − 1+dim H C [( C + r ) ∩ E = α ] > 0 .

28 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?